1 Introduction to the R language
2 Basic data analysis in R
3 Introduction to basic programming in R
4 Probability distributions
5 Statistical Inference
6 Linear models in R
7 Logistic regression
- 7.1 ESR and Plasma Proteins
8 Advanced graphics in R

This course is an introduction to statistical software R. The course will introduce the student to the basics of using R for statistical programming, computation, graphics, and basic statistical modeling.

Objectives:

Use R for basic data analysis.
Write functions in and use R in efficient way,
Perform basic statistical analysis and basic statistical models.

Pre-requisites:

There are no formal prerequisites, but some basic knowledge of statistics is expected. The intended audience is anyone who needs a flexible statistical environment for their research. Any prior knowledge of R is not expected.

Compulsory:

Bring your laptop with the latest version of R installed. Download it here
Rstudio is a user-friendly interface. Download it here Highly recommended!!!
Internet connection to download the material and install some packages.

1 Introduction to the `R` language

R is a free software environment for statistical computing and graphics http://www.r-project.org
R is a command-driven statistical package.
The most important reasons to use R are:
- R is free and multiplatform (Windows/Linux/MACos)
- R allows you to do all the statistical tests/models/analysis you need :)
- Excellent graphics and programming capabilities
- Growing community of users and developers
- Lots of online resources
Statistical features:
- Graphical Techniques (Exploratory Data Analysis)
- Linear and non-linear modeling (linear regression, non-parametric regression, smoothing, etc …)
- Classical statistical tests
- Time-series analysis
- Econometrics
- Survival analysis
- Classification and clustering (data mining, machine learning)
- Optimization and Mathematical Programming
- Bayesian inference etc ….

Visit http://cran.r-project.org/web/views or http://stackoverflow.com/questions/tagged/r

1.1 Start with `R`

Get current working directory

getwd()

list the objects in the current workspace

ls()

Set working directory

setwd("/Users/dlee")

work with your previous commands

history() # display last 25 commands
history(max.show=Inf) # display all previous commands

save your command history

savehistory(file="myfile") # default is ".Rhistory"

recall your command history

loadhistory(file="myfile") # default is ".Rhistory"

save the workspace to the file .RData

save.image()

save specific objects to a file if you don’t specify the path, the cwd is assumed

save(<object list>,file="myfile.RData")

load a workspace into the current session

load("myfile.RData")

quit R. You will be prompted to save the workspace.

q()

1.2 Install and load an `R` library

install.packages("DAAG") # (Data Analysis And Graphics)

Once installed the package, load it

library(DAAG) # or require(DAAG)

1.3 Reading data

The R console

x <- c(7.82,8.00,7.95) # c means "combine"
x

## [1] 7.82 8.00 7.95

A quicker way is to use scan()

x <- scan()  # enter a number followed by return and blank line to end
1: 7.82
2: 8.00
3: 7.95
4: 
Read 3 items

To create a character vector use ""

id <- c("John","Paul","George","Ringo")

To read a character vector

id <- scan(,"")
1: John
2: Paul
3: George
4: Ringo
5: 
Read 4 items

id

## [1] "John"   "Paul"   "George" "Ringo"

1.4 Data Import

In most situations, we need to read data from a separate data file. There are several methods for doing this.

scan() (see ?scan for help)

cat("Example:", "2 3 5 7", "11 13 17", file = "ex.txt", sep = "\n") # creates ex.txt
scan("ex.txt", skip = 1)

## [1]  2  3  5  7 11 13 17

scan("ex.txt", skip = 1, nlines = 1) # only 1 line after the skipped one

## [1] 2 3 5 7

unlink("ex.data") # tidy up

Several formats are available (.txt, .csv, .xls, .xlsx, SAS, Stata, etc…)
Some R libraries to import data are

library(gdata)
library(foreign)

Read data from a .txt or .csv files

mydata1 = read.table("mydata.txt") 
mydata2 = read.csv("mydata.csv")

Other formats .xls and .xlsx

# read in the worksheet named mysheet
mydata <- read.xlsx("myexcel.xlsx", sheetName = "mysheet")

Minitab, SPSS, SAS or Stata

library(foreign)                   
mydata = read.mtp("mydata.mtp")  # Minitab
mydata = read.spss("myfile", to.data.frame=TRUE) # SPSS
mydata = read.dta("mydata.dta") # Stata

library(Hmisc)
mydata = spss.get("mydata.por", use.value.labels=TRUE)  # SPSS

1.5 Exporting data

There are numerous methods for exporting R objects into other formats . For SPSS, SAS and Stata. you will need to load the foreign packages. For Excel, you will need the xlsx package.
Tab delimited text file

mtcars
?mtcars    
write.table(mtcars, "mydata.txt", sep="\t")

Excel spreadsheet

library(xlsx)
write.xlsx(mydata, "mydata.xlsx")

1.6 Data vectors

Download R code here
Create a vector of weights and heights

weight<-c(60,72,57,90,95,72)  
class(weight)

## [1] "numeric"

height<-c(1.75,1.80,1.65,1.90,1.74,1.91)

calculate Body Mass Index

bmi<- weight/height^2
bmi

## [1] 19.59184 22.22222 20.93664 24.93075 31.37799 19.73630

1.7 Basic statistics

mean, median, st dev, variance

mean(weight) 
median(weight)
sd(weight)
var(weight)

summarize data

summary(weight)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   57.00   63.00   72.00   74.33   85.50   95.00

min(weight)
max(weight)
range(weight)
sum(weight)
length(weight)

Quantiles and percentile

There are several quartiles of an observation variable. The first quartile, or lower quartile, is the value that cuts off the first 25% of the data when it is sorted in ascending order. The second quartile, or median, is the value that cuts off the first 50%. The third quartile, or upper quartile, is the value that cuts off the first 75%.

quantile(weight)

##   0%  25%  50%  75% 100% 
## 57.0 63.0 72.0 85.5 95.0

The $n^{\rm th}$ percentile of an observation variable is the value that cuts off the first $n$ percent of the data values when it is sorted in ascending order.

quantile(weight,c(0.32,0.57,0.98))

##  32%  57%  98% 
## 67.2 72.0 94.5

Covariance and correlation

The covariance of two variables $x$ and $y$ in a data sample measures how the two are linearly related. A positive covariance would indicates a positive linear relationship between the variables, and a negative covariance would indicate the opposite.

\[ \rm{Cov}(x,y) = \frac{1}{n}\sum_{i=1}^{n}(x_i - \bar{x})(y_i - \bar{y}) \]

cov(weight,height)

## [1] 0.6773333

The correlation coefficient of two variables in a data sample is their covariance divided by the product of their individual standard deviations. It is a normalized measurement of how the two are linearly related.

Formally, the sample correlation coefficient is defined by the following formula, where $\sigma_x$ and $\sigma_y$ are the sample standard deviations, and $\sigma_xy$ is the covariance.

\[ \rho_{xy} = \frac{\sigma_{xy}}{\sigma_x~\sigma_y} \]

cor(weight,height)

## [1] 0.437934

1.8 Character vectors and factor variables

subject <- c("John","Peter","Chris","Tony","Mary","Jane")
sex <- c("MALE","MALE","MALE","MALE","FEMALE","FEMALE")
class(subject)

## [1] "character"

table(sex)

## sex
## FEMALE   MALE 
##      2      4

1.9 Data frames

Dat <- data.frame(subject,sex,weight,height)
# add bmi to Dat
Dat$bmi <- bmi  # or Dat$bmi <- weight/height^2
class(Dat)

## [1] "data.frame"

str(Dat) # display object structure

## 'data.frame':    6 obs. of  5 variables:
##  $ subject: Factor w/ 6 levels "Chris","Jane",..: 3 5 1 6 4 2
##  $ sex    : Factor w/ 2 levels "FEMALE","MALE": 2 2 2 2 1 1
##  $ weight : num  60 72 57 90 95 72
##  $ height : num  1.75 1.8 1.65 1.9 1.74 1.91
##  $ bmi    : num  19.6 22.2 20.9 24.9 31.4 ...

# Change rownames
rownames(Dat)<-c("A","B","C","D","E","F")

# Access to data frame elements (similar to a matrix)
Dat[,1]     # 1st column

## [1] John  Peter Chris Tony  Mary  Jane 
## Levels: Chris Jane John Mary Peter Tony

Dat[,1:3]   # 1st to 3rd columns

##   subject    sex weight
## A    John   MALE     60
## B   Peter   MALE     72
## C   Chris   MALE     57
## D    Tony   MALE     90
## E    Mary FEMALE     95
## F    Jane FEMALE     72

Dat[1:2,]   # 1st to 2nd row

##   subject  sex weight height      bmi
## A    John MALE     60   1.75 19.59184
## B   Peter MALE     72   1.80 22.22222

1.10 Working with data frames

Example: Analyze data by groups

Obtain the mean weight, heigth and bmi means by FEMALES and MALES:

Select each group and compute the mean

Dat[sex=="MALE",]
Dat[sex=="FEMALE",]

mean(Dat[sex=="MALE",3])  # weight average of MALEs
mean(Dat[sex=="MALE","weight"])

Use apply by columns

apply(Dat[sex=="FEMALE",3:5],2,mean)
apply(Dat[sex=="MALE",3:5],2,mean)

# we can use apply with our own function
apply(Dat[sex=="FEMALE",3:5],2,function(x){x+2})

by and colMeans

# 'by' splits your data by factors and do calculations on each subset.
by(Dat[,3:5],sex, colMeans)

aggregate

# another option
aggregate(Dat[,3:5], by=list(sex),mean)

1.11 Logical vectors

Choose individuals with BMI>22

bmi
bmi>22
as.numeric(bmi>22) # convert a logical condition to a numeric value 0/1
which(bmi>22)  # gives the position of bmi for which bmi>22

Which are between 20 and 25?

bmi > 20 & bmi < 25
which(bmi > 20 & bmi < 25)

1.12 Working with vectors

Concatenate

x <- c(2, 3, 5, 2, 7, 1)
y <- c(10, 15, 12)
z <- c(x,y)  # concatenates x and y

list two vectors

zz <- list(x,y) # create a list
unlist(zz) # unlist the list converting it to a concatenated vector

## [1]  2  3  5  2  7  1 10 15 12

subset of vectors

x[c(1,3,4)]

## [1] 2 5 2

x[-c(2,6)] # negative subscripts omit the chosen elements

## [1] 2 5 2 7

Sequences

seq(1,9) # or 1:9

## [1] 1 2 3 4 5 6 7 8 9

seq(1,9,by=1)

## [1] 1 2 3 4 5 6 7 8 9

seq(1,9,by=0.5)

##  [1] 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0

seq(1,9,length=20)

##  [1] 1.000000 1.421053 1.842105 2.263158 2.684211 3.105263 3.526316
##  [8] 3.947368 4.368421 4.789474 5.210526 5.631579 6.052632 6.473684
## [15] 6.894737 7.315789 7.736842 8.157895 8.578947 9.000000

Replicates

oops <- c(7,9,13)
rep(oops,3) # repeats the entire vector "oops" three times
rep(oops,1:3) # this function has the number 3 replaced 
              #  by a vector with the three values (1,2,3) 
              #  indicating that 7 should be repeated once, 9 twice and 13 three times.

rep(c(2,3,5), 4)
rep(1:2,c(10,15))

rep(c("MALE","FEMALE"),c(4,2)) # it also works with character vectors 
c(rep("MALE",3), rep("FEMALE",2))

1.13 Matrices and arrays

x<- 1:12
x

##  [1]  1  2  3  4  5  6  7  8  9 10 11 12

dim(x)<-c(3,4)  # 3 rows and 4 columns

X <- matrix(1:12,nrow=3,byrow=TRUE)
X

##      [,1] [,2] [,3] [,4]
## [1,]    1    2    3    4
## [2,]    5    6    7    8
## [3,]    9   10   11   12

X <- matrix(1:12,nrow=3,byrow=FALSE)
X

##      [,1] [,2] [,3] [,4]
## [1,]    1    4    7   10
## [2,]    2    5    8   11
## [3,]    3    6    9   12

# rownames, colnames

rownames(X) <- c("A","B","C")
X

##   [,1] [,2] [,3] [,4]
## A    1    4    7   10
## B    2    5    8   11
## C    3    6    9   12

colnames(X) <- LETTERS[4:7]
X

##   D E F  G
## A 1 4 7 10
## B 2 5 8 11
## C 3 6 9 12

colnames(X) <- month.abb[4:7]
X

##   Apr May Jun Jul
## A   1   4   7  10
## B   2   5   8  11
## C   3   6   9  12

Column/Row bind operations cbind(), rbind()

Y <- matrix(0.1*(1:12),3,4)

cbind(X,Y)  # bind column-wise

##   Apr May Jun Jul                
## A   1   4   7  10 0.1 0.4 0.7 1.0
## B   2   5   8  11 0.2 0.5 0.8 1.1
## C   3   6   9  12 0.3 0.6 0.9 1.2

rbind(X,Y)  # bind row-wise

##   Apr May Jun  Jul
## A 1.0 4.0 7.0 10.0
## B 2.0 5.0 8.0 11.0
## C 3.0 6.0 9.0 12.0
##   0.1 0.4 0.7  1.0
##   0.2 0.5 0.8  1.1
##   0.3 0.6 0.9  1.2

1.14 Factors

gender<-c(rep("female",691),rep("male",692))
class(gender)

## [1] "character"

# change vector to factor (i.e. a category)
gender<- factor(gender)
levels(gender)

## [1] "female" "male"

summary(gender)

## female   male 
##    691    692

table(gender)

## gender
## female   male 
##    691    692

status<- c(0,3,2,1,4,5)    # This command creates a numerical vector pain, 
                           #    encoding the pain level of five patients.
fstatus <- factor(status, levels=0:5)
levels(fstatus) <- c("student","engineer","unemployed","lawyer","economist","dentist")

Dat$status <- fstatus
Dat

##   subject    sex weight height      bmi     status
## A    John   MALE     60   1.75 19.59184    student
## B   Peter   MALE     72   1.80 22.22222     lawyer
## C   Chris   MALE     57   1.65 20.93664 unemployed
## D    Tony   MALE     90   1.90 24.93075   engineer
## E    Mary FEMALE     95   1.74 31.37799  economist
## F    Jane FEMALE     72   1.91 19.73630    dentist

1.15 Indexing vector with logicals

a <- c(1,2,3,4,5)
b <- c(TRUE,FALSE,FALSE,TRUE,FALSE)

max(a[b])

## [1] 4

sum(a[b])

## [1] 5

1.16 Missing values (NA)

a <- c(1,2,3,4,NA)
sum(a)

## [1] NA

sum(a,na.rm=TRUE)

## [1] 10

a <- c(1,2,3,4,NA)
is.na(a)

## [1] FALSE FALSE FALSE FALSE  TRUE

1.17 Working with data frames

A data frame is used for storing data tables. It is a list of vectors of equal length.

mtcars
?mtcars       # or help(mtcars)

look at the first rows

head(mtcars)

##                    mpg cyl disp  hp drat    wt  qsec vs am gear carb
## Mazda RX4         21.0   6  160 110 3.90 2.620 16.46  0  1    4    4
## Mazda RX4 Wag     21.0   6  160 110 3.90 2.875 17.02  0  1    4    4
## Datsun 710        22.8   4  108  93 3.85 2.320 18.61  1  1    4    1
## Hornet 4 Drive    21.4   6  258 110 3.08 3.215 19.44  1  0    3    1
## Hornet Sportabout 18.7   8  360 175 3.15 3.440 17.02  0  0    3    2
## Valiant           18.1   6  225 105 2.76 3.460 20.22  1  0    3    1

Structure of the data frame

str(mtcars) # display the structure of the data frame

## 'data.frame':    32 obs. of  11 variables:
##  $ mpg : num  21 21 22.8 21.4 18.7 18.1 14.3 24.4 22.8 19.2 ...
##  $ cyl : num  6 6 4 6 8 6 8 4 4 6 ...
##  $ disp: num  160 160 108 258 360 ...
##  $ hp  : num  110 110 93 110 175 105 245 62 95 123 ...
##  $ drat: num  3.9 3.9 3.85 3.08 3.15 2.76 3.21 3.69 3.92 3.92 ...
##  $ wt  : num  2.62 2.88 2.32 3.21 3.44 ...
##  $ qsec: num  16.5 17 18.6 19.4 17 ...
##  $ vs  : num  0 0 1 1 0 1 0 1 1 1 ...
##  $ am  : num  1 1 1 0 0 0 0 0 0 0 ...
##  $ gear: num  4 4 4 3 3 3 3 4 4 4 ...
##  $ carb: num  4 4 1 1 2 1 4 2 2 4 ...

Select a car model:

mtcars["Mazda RX4",] # using rows and columns names

##           mpg cyl disp  hp drat   wt  qsec vs am gear carb
## Mazda RX4  21   6  160 110  3.9 2.62 16.46  0  1    4    4

mtcars[c("Datsun 710", "Camaro Z28"),]

##             mpg cyl disp  hp drat   wt  qsec vs am gear carb
## Datsun 710 22.8   4  108  93 3.85 2.32 18.61  1  1    4    1
## Camaro Z28 13.3   8  350 245 3.73 3.84 15.41  0  0    3    4

Or specific variables

mtcars[,c("mpg","am")]

##                      mpg am
## Mazda RX4           21.0  1
## Mazda RX4 Wag       21.0  1
## Datsun 710          22.8  1
## Hornet 4 Drive      21.4  0
## Hornet Sportabout   18.7  0
## Valiant             18.1  0
## Duster 360          14.3  0
## Merc 240D           24.4  0
## Merc 230            22.8  0
## Merc 280            19.2  0
## Merc 280C           17.8  0
## Merc 450SE          16.4  0
## Merc 450SL          17.3  0
## Merc 450SLC         15.2  0
## Cadillac Fleetwood  10.4  0
## Lincoln Continental 10.4  0
## Chrysler Imperial   14.7  0
## Fiat 128            32.4  1
## Honda Civic         30.4  1
## Toyota Corolla      33.9  1
## Toyota Corona       21.5  0
## Dodge Challenger    15.5  0
## AMC Javelin         15.2  0
## Camaro Z28          13.3  0
## Pontiac Firebird    19.2  0
## Fiat X1-9           27.3  1
## Porsche 914-2       26.0  1
## Lotus Europa        30.4  1
## Ford Pantera L      15.8  1
## Ferrari Dino        19.7  1
## Maserati Bora       15.0  1
## Volvo 142E          21.4  1

library(psych)
describe(mtcars)

##      vars  n   mean     sd median trimmed    mad   min    max  range  skew
## mpg     1 32  20.09   6.03  19.20   19.70   5.41 10.40  33.90  23.50  0.61
## cyl     2 32   6.19   1.79   6.00    6.23   2.97  4.00   8.00   4.00 -0.17
## disp    3 32 230.72 123.94 196.30  222.52 140.48 71.10 472.00 400.90  0.38
## hp      4 32 146.69  68.56 123.00  141.19  77.10 52.00 335.00 283.00  0.73
## drat    5 32   3.60   0.53   3.70    3.58   0.70  2.76   4.93   2.17  0.27
## wt      6 32   3.22   0.98   3.33    3.15   0.77  1.51   5.42   3.91  0.42
## qsec    7 32  17.85   1.79  17.71   17.83   1.42 14.50  22.90   8.40  0.37
## vs      8 32   0.44   0.50   0.00    0.42   0.00  0.00   1.00   1.00  0.24
## am      9 32   0.41   0.50   0.00    0.38   0.00  0.00   1.00   1.00  0.36
## gear   10 32   3.69   0.74   4.00    3.62   1.48  3.00   5.00   2.00  0.53
## carb   11 32   2.81   1.62   2.00    2.65   1.48  1.00   8.00   7.00  1.05
##      kurtosis    se
## mpg     -0.37  1.07
## cyl     -1.76  0.32
## disp    -1.21 21.91
## hp      -0.14 12.12
## drat    -0.71  0.09
## wt      -0.02  0.17
## qsec     0.34  0.32
## vs      -2.00  0.09
## am      -1.92  0.09
## gear    -1.07  0.13
## carb     1.26  0.29

2 Basic data analysis in `R`

2.1 Basic plotting

Scatterplot

attach(mtcars)
plot(wt, mpg, main="Scatterplot Example",
   xlab="Car Weight ", ylab="Miles Per Gallon ", pch=19)

Basic Scatterplot Matrix

pairs(~mpg+disp+drat+wt,data=mtcars,
   main="Simple Scatterplot Matrix")

Barplot

tab <- table(mtcars[,c("cyl")])
barplot(tab)

Piechart

pie(tab)

Exercises:

The data.frame VADeaths contains the death rates per 1000 in Virginia (US) in 1940

The death rates are measured per 1000 population per year. They are cross-classified by age group (rows) and population group (columns). The age groups are: 50-54, 55-59, 60-64, 65-69, 70-74 and the population groups are Rural/Male, Rural/Female, Urban/Male and Urban/Female.

data(VADeaths)
VADeaths

##       Rural Male Rural Female Urban Male Urban Female
## 50-54       11.7          8.7       15.4          8.4
## 55-59       18.1         11.7       24.3         13.6
## 60-64       26.9         20.3       37.0         19.3
## 65-69       41.0         30.9       54.6         35.1
## 70-74       66.0         54.3       71.1         50.0

Compute the mean for each age group.
- Result:

##  50-54  55-59  60-64  65-69  70-74 
## 11.050 16.925 25.875 40.400 60.350

Compute the mean for each population group.
- Result:

##   Rural Male Rural Female   Urban Male Urban Female 
##        32.74        25.18        40.48        25.28

The data.frame rainforest contains several variables from different species

library(DAAG)
rainforest

Create a table of counts for each species and make a graphic with the results.
- Result:

## 
## Acacia mabellae      C. fraseri  Acmena smithii   B. myrtifolia 
##              16              12              26              11

The Acmena data.frame is created from rainforest using the function subset.

Plot the relationship between the wood biomass (wood) and the diameter of the breast height (dbh). Use also a logarithm scale.

Acmena <- subset(rainforest, species == "Acmena smithii")

Compute a histogram of variable dbh using function hist

Create a vector of the positive odd integers less than 100 and remove the values greater than 60 and less than 80.
- Result:

##  [1] 61 63 65 67 69 71 73 75 77 79

Solutions here

2.2 Scatterplots

library(MASS)
data("mammals")
?mammals
head(mammals)

##                    body brain
## Arctic fox        3.385  44.5
## Owl monkey        0.480  15.5
## Mountain beaver   1.350   8.1
## Cow             465.000 423.0
## Grey wolf        36.330 119.5
## Goat             27.660 115.0

attach(mammals)
species <- row.names(mammals)
x <- body
y <- brain

library(calibrate)
# scatterplot
plot(x,y, xlab = "body weight in kgr", ylab = "brain weight in gr", 
     main="Body vs Brain weight \n for 62 Species of Land Mammals")
textxy(x,y,labs=species,col = "blue",cex=0.85)

Identify a point in the scatterplot

identify(x,y,species)

Plot in the log scale

plot(log(x),log(y), xlab = "log body weight in kgr", ylab = "log brain weight in gr", 
     main="log Body vs log Brain weight \n for 62 Species of Land Mammals")
textxy(log(x),log(y),labs=species,col = "blue",cex=0.85)

Identify a point in the log scale scatterplot

identify(log(x),log(y),species)

2.3 More plotting options

Multiple Data Sets on One Plot

One common task is to plot multiple data sets on the same plot. In many situations the way to do this is to create the initial plot and then add additional information to the plot. For example, to plot bivariate data the plot command is used to initialize and create the plot. The points command can then be used to add additional data sets to the plot.

 x <- rnorm(10,sd=5,mean=20)
 y <- 2.5*x - 1.0 + rnorm(10,sd=9,mean=0)
 cor(x,y)

## [1] 0.6412437

 plot(x,y,xlab="Independent",ylab="Dependent",main="Random plot")
 x1 <- runif(8,15,25)
 y1 <- 2.5*x1 - 1.0 + runif(8,-6,6)
 points(x1,y1,col=2)

with legend and $(x_2,y_2)$ points:

x2 <- runif(8,15,25)
y2 <- 2.5*x2 - 1.0 + runif(8,-6,6)
 plot(x,y,xlab="Independent",ylab="Dependent",main="Random plot")
 points(x1,y1,col=2,pch=3)
 points(x2,y2,col=4,pch=5)
 legend("topleft",c("Original","one","two"),col=c(1,2,4),pch=c(1,3,5))

Errors bars:

plot(x,y,xlab="Independent",ylab="Dependent",main="Random plot",ylim=c(20,90))
xHigh <- x
yHigh <- y + abs(rnorm(10,sd=3.5))
xLow <- x
yLow <- y - abs(rnorm(10,sd=3.1))
arrows(xHigh,yHigh,xLow,yLow,col=2,angle=90,length=0.1,code=3)

plot(1:20,0*(1:20),pch=1:20,cex=2)

Multiple Graphs on One Image:

 par(mfrow=c(2,3))
 boxplot(rnorm(100),main="first plot")
 boxplot(rgamma(100,2),main="second plot", horizontal=TRUE,col="bisque")
 plot(rnorm(100),xlab="third plot",
      ylab="y-label",main="x-label")
 hist(rnorm(100),main="fourth plot",col="lightgrey")
 hist(rexp(100),main="fifth plot",col="blue")
 plot(rnorm(100),rexp(100),main="sixth plot")

Pairwise relationships

uData <- rnorm(20)
vData <- rnorm(20,mean=5)
wData <- uData + 2*vData + rnorm(20,sd=0.5)
xData <- -2*uData+rnorm(20,sd=0.1)
yData <- 3*vData+rnorm(20,sd=2.5)
d <- data.frame(u=uData,v=vData,w=wData,x=xData,y=yData)
pairs(d)

Plotting correlations

The function corrplot in the library(corrplot) visualizes a correlation matrix calculate with function cor

library(corrplot)
M <- cor(d)
corrplot(M, method="circle",type="upper")

Plotting surfaces: image, contour and persp plots

x <- seq(0,2*pi,by=pi/50)
y <- x
xg <- (x*0+1) %*% t(y)
yg <- (x) %*% t(y*0+1)
f <- sin(xg*yg)

par(mfrow=c(2,2))
image(x,y,f)
contour(x,y,f)
contour(x,y,f,nlevels=4)
image(x,y,f,col=grey.colors(100))
contour(x,y,f,nlevels=4,add=TRUE,col="red")

Similarly, one can use persp plot

persp(x,y,f,theta=-30,phi=55,col="lightgrey",shade=.01)

2.4 QQ-plot

A Q-Q plot is a plot of the quantiles of two distributions against each other, or a plot based on estimates of the quantiles. The pattern of points in the plot is used to compare the two distributions.

qqnorm is used to determine if your data is close to being normally distributed. You cannot be sure that the data is normally distributed, but you can rule out if it is not normally distributed.

set.seed(1234)

require(graphics)
y <- rt(200, df = 5)
qqnorm(y); 
qqline(y, col = 2)

z <- rnorm(200,mean=0,sd=1)
qqnorm(z)
qqline(z, col = 2)

To compare two distributions

set.seed(1234)
qqplot(qchisq(ppoints(500), df = 3), y,
       main = expression("Q-Q plot for" ~~ {chi^2}[nu == 3]))
qqline(y, distribution = function(p) qchisq(p, df = 3),
       prob = c(0.1, 0.6), col = 2)

If the two distributions being compared are identical, the Q-Q plot follows the $45Âº$ line y = x.

2.5 Tables and Cross-classification

library(MASS)
data(quine)
?quine
attach(quine)
table(Sex)

## Sex
##  F  M 
## 80 66

table(Sex,Age)

##    Age
## Sex F0 F1 F2 F3
##   F 10 32 19 19
##   M 17 14 21 14

# or xtabs
xtabs(~Sex+Age,data=quine)

##    Age
## Sex F0 F1 F2 F3
##   F 10 32 19 19
##   M 17 14 21 14

xtabs(~Sex+Age+Eth,data=quine)

## , , Eth = A
## 
##    Age
## Sex F0 F1 F2 F3
##   F  5 15  9  9
##   M  8  5 11  7
## 
## , , Eth = N
## 
##    Age
## Sex F0 F1 F2 F3
##   F  5 17 10 10
##   M  9  9 10  7

2.6 Calculation of cross-classifications

tapply(Days,Age,mean)

##       F0       F1       F2       F3 
## 14.85185 11.15217 21.05000 19.60606

tapply(Days,list(Sex,Age),mean)

##         F0       F1       F2       F3
## F 18.70000 12.96875 18.42105 14.00000
## M 12.58824  7.00000 23.42857 27.21429

tapply(Days,list(Sex,Age),function(x) sqrt(var(x)/length(x)))

##         F0       F1       F2       F3
## F 4.208589 2.329892 5.299959 2.940939
## M 3.768151 1.418093 3.766122 4.569582

2.7 Qualitative data

A data sample is called qualitative, also known as categorical, if its values belong to a collection of known defined non-overlapping classes. Common examples include student letter grade (A, B, C, D or F), commercial bond rating (AAA, AAB, …) and consumer clothing shoe sizes (1, 2, 3, …).

Let us consider some artificial data consisting of the treatment and improvement of patients with rheumatoid arthritis.

treatment <- factor(rep(c(1, 2), c(43, 41)), levels = c(1, 2),
                    labels = c("placebo", "treated"))
improved <- factor(rep(c(1, 2, 3, 1, 2, 3), c(29, 7, 7, 13, 7, 21)),
                   levels = c(1, 2, 3),
                   labels = c("none", "some", "marked"))

We can compute a cross-classification table

xtabs(~treatment+improved)

##          improved
## treatment none some marked
##   placebo   29    7      7
##   treated   13    7     21

Graphically,

spineplot(improved ~ treatment)

The R dataset UCBAdmissions contains aggregated data on applicants to graduate school at Berkeley for the six largest departments in 1973 classified by admission and sex.

data("UCBAdmissions")
?UCBAdmissions
apply(UCBAdmissions, c(2,1), sum)

##         Admit
## Gender   Admitted Rejected
##   Male       1198     1493
##   Female      557     1278

prop.table(apply(UCBAdmissions, c(2,1), sum))

##         Admit
## Gender    Admitted  Rejected
##   Male   0.2646929 0.3298719
##   Female 0.1230667 0.2823685

ftable(UCBAdmissions)

##                 Dept   A   B   C   D   E   F
## Admit    Gender                             
## Admitted Male        512 353 120 138  53  22
##          Female       89  17 202 131  94  24
## Rejected Male        313 207 205 279 138 351
##          Female       19   8 391 244 299 317

The same but with a more readable format can be obtained using ftable

ftable(round(prop.table(UCBAdmissions), 3),
       row.vars="Dept", col.vars = c("Gender", "Admit"))

##      Gender     Male            Female         
##      Admit  Admitted Rejected Admitted Rejected
## Dept                                           
## A              0.113    0.069    0.020    0.004
## B              0.078    0.046    0.004    0.002
## C              0.027    0.045    0.045    0.086
## D              0.030    0.062    0.029    0.054
## E              0.012    0.030    0.021    0.066
## F              0.005    0.078    0.005    0.070

More interesting are the proportions admitted for each Gender by Dept combination (dimensions 2 and 3 of the array). Notice that male and female admission rates are about the same in all departments, except “A”, where female admission rates are higher.

# prop.table(UCBAdmissions, c(2,3))
ftable(round(prop.table(UCBAdmissions, c(2,3)), 2),
       row.vars="Dept", col.vars = c("Gender", "Admit"))

##      Gender     Male            Female         
##      Admit  Admitted Rejected Admitted Rejected
## Dept                                           
## A               0.62     0.38     0.82     0.18
## B               0.63     0.37     0.68     0.32
## C               0.37     0.63     0.34     0.66
## D               0.33     0.67     0.35     0.65
## E               0.28     0.72     0.24     0.76
## F               0.06     0.94     0.07     0.93

## Data aggregated over departments
apply(UCBAdmissions, c(1, 2), sum)

##           Gender
## Admit      Male Female
##   Admitted 1198    557
##   Rejected 1493   1278

Applications and admissions by department at UC Berkeley can be viewed graphically

spineplot(margin.table(UCBAdmissions, c(3, 2)),
           main = "Applications at UCB")

spineplot(margin.table(UCBAdmissions, c(3, 1)),
           main = "Admissions at UCB")

This data set is frequently used for illustrating Simpson’s paradox. At issue is whether the data show evidence of sex bias in admission practices. There were 2691 male applicants, of whom 1198 (44.5%) were admitted, compared with 1835 female applicants of whom 557 (30.4%) were admitted. Men were much more successful in admissions than women. Wikipedia: Gender Bias UC Berkeley.

2.8 Quantitative data

Quantitative data, also known as continuous data, consists of numeric data that support arithmetic operations. This is in contrast with qualitative data, whose values belong to pre-defined classes with no arithmetic operation allowed. We will explain how to apply some of the R tools for quantitative data analysis with examples.

head(faithful)

##   eruptions waiting
## 1     3.600      79
## 2     1.800      54
## 3     3.333      74
## 4     2.283      62
## 5     4.533      85
## 6     2.883      55

It consists of a collection of observations of the Old Faithful geyser in the USA Yellowstone National Park.

There are two observation variables in the data set. The first one, called eruptions, is the duration of the geyser eruptions. The second one, called waiting, is the length of waiting period until the next eruption. It turns out there is a correlation between the two variables.

plot(faithful)

2.8.1 Frequency distribution of quantitative data

The frequency distribution of a data variable is a summary of the data occurrence in a collection of non-overlapping categories.

Let us find the frequency distribution of the eruption duration in faithful data set.

duration <- faithful$eruptions
range(duration)

## [1] 1.6 5.1

Now we create the range of non-overlapping sub-intervals by defining a sequence of equal distance break points. If we round the endpoints of the interval [1.6, 5.1] to the closest half-integers, we come up with the interval [1.5, 5.5]. Hence we set the break points to be the half-integer sequence { 1.5, 2.0, 2.5, … }.

breaks <- seq(1.5,5.5,by=0.5)
breaks

## [1] 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5

Classify the eruption durations according to the half-unit-length sub-intervals with cut. As the intervals are to be closed on the left, and open on the right, we set the right argument as FALSE.

duration.cut = cut(duration, breaks, right=FALSE)

Compute the frequency of eruptions in each sub-interval with the table function.

duration.freq = table(duration.cut) 
duration.freq

## duration.cut
## [1.5,2) [2,2.5) [2.5,3) [3,3.5) [3.5,4) [4,4.5) [4.5,5) [5,5.5) 
##      51      41       5       7      30      73      61       4

hist function does all the computaions to find the frequency distribution:

freq <- hist(duration)
freq

## $breaks
## [1] 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5
## 
## $counts
## [1] 55 37  5  9 34 75 54  3
## 
## $density
## [1] 0.40441176 0.27205882 0.03676471 0.06617647 0.25000000 0.55147059
## [7] 0.39705882 0.02205882
## 
## $mids
## [1] 1.75 2.25 2.75 3.25 3.75 4.25 4.75 5.25
## 
## $xname
## [1] "duration"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"

freq <- hist(duration,breaks = breaks)

hist(duration,50)

Density estimation builds an estimate of some underlying probability density function using an observed data sample.

require(graphics)
d <- density(faithful$eruptions)
d

## 
## Call:
##  density.default(x = faithful$eruptions)
## 
## Data: faithful$eruptions (272 obs.); Bandwidth 'bw' = 0.3348
## 
##        x                y            
##  Min.   :0.5957   Min.   :0.0002262  
##  1st Qu.:1.9728   1st Qu.:0.0514171  
##  Median :3.3500   Median :0.1447010  
##  Mean   :3.3500   Mean   :0.1813462  
##  3rd Qu.:4.7272   3rd Qu.:0.3086071  
##  Max.   :6.1043   Max.   :0.4842095

plot(d)

Two dimension histogram:

library(gplots)
h2 <- hist2d(faithful, nbins=30,xlab="Duration in minutes",ylab="Waiting")

h2

## 
## ----------------------------
## 2-D Histogram Object
## ----------------------------
## 
## Call: hist2d(x = faithful, nbins = 30, xlab = "Duration in minutes", 
##     ylab = "Waiting")
## 
## Number of data points:  272 
## Number of grid bins:  30 x 30 
## X range: ( 1.6 , 5.1 )
## Y range: ( 43 , 96 )

names(h2)

## [1] "counts"   "x.breaks" "y.breaks" "x"        "y"        "nobs"    
## [7] "call"

Relative frequencies

duration.relfreq <- duration.freq / nrow(faithful) 
tab <- cbind(duration.freq, duration.relfreq) 
apply(tab,2,sum)

##    duration.freq duration.relfreq 
##              272                1

Cumulative frequency distribution

cumsum(duration.freq)

## [1.5,2) [2,2.5) [2.5,3) [3,3.5) [3.5,4) [4,4.5) [4.5,5) [5,5.5) 
##      51      92      97     104     134     207     268     272

cumsum(duration.relfreq)

##   [1.5,2)   [2,2.5)   [2.5,3)   [3,3.5)   [3.5,4)   [4,4.5)   [4.5,5) 
## 0.1875000 0.3382353 0.3566176 0.3823529 0.4926471 0.7610294 0.9852941 
##   [5,5.5) 
## 1.0000000

We can plot the cumulative relative frequency graph of a quantitative variable, which is a curve graphically showing the cumulative relative frequency distribution. The e.c.d.f. (empirical cumulative distribution function) $F_n$ is a step function with jumps $i/n$ at observation values, where $i$ is the number of tied observations at that value. Missing values are ignored.

For observations $x = (x_1,x_2, ... x_n)$, $F_n$ is the fraction of observations less or equal to $t$, i.e.,

\[ F_n(t) = \#{x_i <= t}/n = 1/n \sum_{i=1}^n I(x_i \leq t). \] where $I$ is an indication function.

plot(ecdf(duration))

Bivariante Density estimation:

data("faithful")
attach(faithful)
Dens2d<-kde2d(eruptions,waiting)
image(Dens2d,xlab="eruptions",ylab="waiting")
contour(Dens2d,add=TRUE,col="black",lwd=2,nlevels=5)

detach("faithful")

Perspective plot:

persp(Dens2d,phi=30,theta=20,d=5,xlab="eruptions",ylab="waiting",zlab="",shade=.2,col="lightblue",expand=.85,ticktype = "detailed")

3 Introduction to basic programming in `R`

3.1 Control Structures

3.1.1 Conditional Executions

3.1.1.1 Comparison Operators

equal: ==

  "hola" == "hola"

## [1] TRUE

  "hola" == "Hola"

## [1] FALSE

   1 == 2-1

## [1] TRUE

not equal: !=

    a <- c(1,2,4,5)
    b <- c(1,2,3,5) 
    a == b

## [1]  TRUE  TRUE FALSE  TRUE

    a != b

## [1] FALSE FALSE  TRUE FALSE

greater/less than: > <

set.seed(1)
a <- rnorm(10,3,1)
b <- rnorm(10,4,2)
a<b

##  [1]  TRUE  TRUE  TRUE FALSE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE

greater/less than or equal: >= <=

set.seed(1)
a <- rpois(10,1)
b <- rbinom(10,2,.56)
a >= b

##  [1] FALSE FALSE  TRUE  TRUE FALSE  TRUE  TRUE  TRUE  TRUE FALSE

which

set.seed(1)
which(a>b)

## [1] 4 6 7 8

LETTERS

##  [1] "A" "B" "C" "D" "E" "F" "G" "H" "I" "J" "K" "L" "M" "N" "O" "P" "Q"
## [18] "R" "S" "T" "U" "V" "W" "X" "Y" "Z"

which(LETTERS=="R")

## [1] 18

which.min or which.max

set.seed(1)
a <- rnorm(10,2,1)
a

##  [1] 1.373546 2.183643 1.164371 3.595281 2.329508 1.179532 2.487429
##  [8] 2.738325 2.575781 1.694612

which.min(a)

## [1] 3

which.max(a)

## [1] 4

is.na

 a[2] <- NA
is.na(a)

##  [1] FALSE  TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE

which(is.na(a))

## [1] 2

3.1.2 Logical Operators

and: &

z = 1:6
which(2 < z & z > 3)

## [1] 4 5 6

or: |

z = 1:6
(z > 2) & (z < 5)

## [1] FALSE FALSE  TRUE  TRUE FALSE FALSE

which((z > 2) & (z < 5))

## [1] 3 4

not: !

x <- c(TRUE,FALSE,0,6)
y <- c(FALSE,TRUE,FALSE,TRUE)

!x

## [1] FALSE  TRUE  TRUE FALSE

Operators & and | perform element-wise operation producing result having length of the longer operand. But && and || examines only the first element of the operands resulting into a single length logical vector. Zero is considered FALSE and non-zero numbers are taken as TRUE. Example:

&& vs &

x&y

## [1] FALSE FALSE FALSE  TRUE

x&&y

## [1] FALSE

|| vs |

x||y

## [1] TRUE

x|y

## [1]  TRUE  TRUE FALSE  TRUE

3.2 `if` statements

if(cond1=true) { cmd1 } else { cmd2 }

if(1==0) {
    print(1)
} else {
    print(2)
}

## [1] 2

3.3 `ifelse` statement

ifelse(test, true_value, false_value)

x <- 1:10 # Creates sample data
ifelse(x<5 | x>8, x, 0)

##  [1]  1  2  3  4  0  0  0  0  9 10

3.4 `while` statement

3.5 Loops

The most commonly used loop structures in R are for, while and apply loops. Less common are repeat loops. The break function is used to break out of loops, and next halts the processing of the current iteration and advances the looping index.

3.5.1 `for`

For loops are controlled by a looping vector. In every iteration of the loop one value in the looping vector is assigned to a variable that can be used in the statements of the body of the loop. Usually, the number of loop iterations is defined by the number of values stored in the looping vector and they are processed in the same order as they are stored in the looping vector.

Syntax

for(variable in sequence) {
    statements
}

for (j in 1:5)
{
  print(j^2)
}

## [1] 1
## [1] 4
## [1] 9
## [1] 16
## [1] 25

Repeat the loop saving the resuls in a vector x.

n = 5
x = NULL  # creates a NULL object
for (j in 1:n)
{
  x[j] = j^2
}
x

## [1]  1  4  9 16 25

Let’s use a for loop to estimate the average of squaring the result of a roll of a dice.

nsides = 6
ntrials = 1000
trials = NULL
for (j in 1:ntrials)
{
  trials[j] = sample(1:nsides,1)  # We get one sample at a time
}
mean(trials^2)

## [1] 15.122

Example: stop on condition and print error message

x <- 1:10
z <- NULL
for(i in seq(along=x)) {
    if (x[i]<5) {
        z <- c(z,x[i]-1) 
    } else {
        stop("values need to be <5")
    }
}
## Error: values need to be <5
z
## [1] 0 1 2 3

3.6 `while`

Similar to for loop, but the iterations are controlled by a conditional statement.

z <- 0
while(z < 5) {
    z <- z + 2
    print(z) 
}

## [1] 2
## [1] 4
## [1] 6

3.7 `apply` loop family

For Two-Dimensional Data Sets: apply

Syntax:

apply(X, MARGIN, FUN, ARGs)

X: array, matrix or data.frame; MARGIN: 1 for rows, 2 for columns, c(1,2) for both; FUN: one or more functions; ARGs: possible arguments for function.

## Example for applying predefined mean function
apply(mtcars[,1:3], 1, mean)

## With custom function
x <- 1:10
test <- function(x) { # Defines some custom function
    if(x < 5) {
        x-1
    } else {
        x / x
    }
} 

apply(as.matrix(x), 1, test) 

## Same as above but with a single line of code
apply(as.matrix(x), 1, function(x) { if (x<5) { x-1 } else { x/x } })

For Ragged Arrays: tapply

Apply a function to each cell of a ragged array, that is to each (non-empty) group of values given by a unique combination of the levels of certain factors.

## Computes mean values of vector agregates defined by factor
tapply(as.vector(mtcars$mpg), factor(mtcars$cyl), mean)

##        4        6        8 
## 26.66364 19.74286 15.10000

## The aggregate function provides related utilities
aggregate(mtcars[,c(1,3,4)], list(mtcars$cyl), mean)

##   Group.1      mpg     disp        hp
## 1       4 26.66364 105.1364  82.63636
## 2       6 19.74286 183.3143 122.28571
## 3       8 15.10000 353.1000 209.21429

For Vectors and Lists: lapply and sapply

Both apply a function to vector or list objects. The function lapply returns a list, while sapply attempts to return the simplest data object, such as vector or matrix instead of list.

Syntax

lapply(X,FUN)
sapply(X,FUN)

## $mpg
##  [1] 21.0 21.0 22.8 21.4 18.7 18.1 14.3 24.4 22.8 19.2 17.8 16.4 17.3 15.2
## [15] 10.4 10.4 14.7 32.4 30.4 33.9 21.5 15.5 15.2 13.3 19.2 27.3 26.0 30.4
## [29] 15.8 19.7 15.0 21.4
## 
## $hp
##  [1] 110 110  93 110 175 105 245  62  95 123 123 180 180 180 205 215 230
## [18]  66  52  65  97 150 150 245 175  66  91 113 264 175 335 109
## 
## $wt
##  [1] 2.620 2.875 2.320 3.215 3.440 3.460 3.570 3.190 3.150 3.440 3.440
## [12] 4.070 3.730 3.780 5.250 5.424 5.345 2.200 1.615 1.835 2.465 3.520
## [23] 3.435 3.840 3.845 1.935 2.140 1.513 3.170 2.770 3.570 2.780

Compute sum of each list component and return result as list

lapply(mylist, sum)

## $mpg
## [1] 642.9
## 
## $hp
## [1] 4694
## 
## $wt
## [1] 102.952

Compute sum of each list component and return result as vector

sapply(mylist, sum)

##      mpg       hp       wt 
##  642.900 4694.000  102.952

3.8 Other Loops

Repeat Loop

Syntax

repeat statements

Loop is repeated until a break is specified. This means there needs to be a second statement to test whether or not to break from the loop.

Example:

z <- 0
repeat {
    z <- z + 1
    print(z)
    if(z > 100) break()
}

3.9 Improving Speed Performance of Loops

Looping over very large data sets can become slow in R. However, this limitation can be overcome by eliminating certain operations in loops or avoiding loops over the data intensive dimension in an object altogether. The latter can be achieved by performing mainly vector-to-vector or matrix-to-matrix computations which run often over 100 times faster than the corresponding for() or apply() loops in R. For this purpose, one can make use of the existing speed-optimized R functions (e.g.: rowSums, rowMeans, table, tabulate) or one can design custom functions that avoid expensive R loops by using vector- or matrix-based approaches. Alternatively, one can write programs that will perform all time consuming computations on the C-level.

Speed comparison of for loops with an append versus and inject step

N <- 1e6
myMA <- matrix(rnorm(N), N, 10, dimnames=list(1:N, paste("C", 1:10, sep="")))
results <- NULL
system.time(for(i in seq(along=myMA[,1])) 
            results <- c(results, mean(myMA[i,])))

results <- numeric(length(myMA[,1]))
system.time(for(i in seq(along=myMA[,1])) 
            results[i] <- mean(myMA[i,]))

The inject approach is 20-50 times faster than the append version.

Speed comparison of apply loop versus rowMeans for computing the mean for each row in a large matrix:

system.time(myMAmean <- apply(myMA, 1, mean))
system.time(myMAmean <- rowMeans(myMA))

The rowMeans approach is over 200 times faster than the apply loop.

4 Probability distributions

4.1 Binomial distribution $Bin(n,p)$

The binomial distribution is a discrete probability distribution. It describes the outcome of $n$ independent trials in an experiment. Each trial is assumed to have only two outcomes, either success or failure. If the probability of a successful trial is $p$, then the probability of having $k$ successful outcomes in an experiment of $n$ independent trials is given by the probability mass function:.

\[ f(k,n,p) = \mbox{Pr}(X=k)=\binom{n}{k} p^k (1-p)^{n-k}, \quad k=0,1,2,...,n \]

The cumulative distribution function can be expressed as:

\[ F(k;n,p) = \mbox{Pr}(X\leq k) = \sum_{i=0}^{k}\binom{n}{i} p^i (1-p)^{n-i} \]

with mean $np$ and variance $np(1-p)$.

Question:

Suppose there are twelve multiple choice questions in an Maths class quiz. Each question has five possible answers, and only one of them is correct. Find the probability of having four or less correct answers if a student attempts to answer every question at random.

What is the probability of 2 or 3 questions answered correctly?

Question:

Suppose company A manufactures a product B which have probability 0.005 of being defective. Suppose product B is shipped in cartons containing 25 B items. What is the probability that a randomly chosen carton contains exactly one defective product? What is the probability that a randomly chosen carton contains no more than one defective widgit?

Solutions here

4.2 Poisson distribution $Pois(\lambda)$

The Poisson distribution is the probability distribution of independent event occurrences in an interval. If $\lambda$ is the mean occurrence per interval, then the probability of having $k$ occurrences within a given interval is the probability mass function given by:

\[ \mbox{Pr}(\mbox{$k$ events in interval}) = \frac{\lambda^k e^{-\lambda}}{k!} \]

The cumulative density function for the Poisson cumulative probability function is \[ P(X\leq x ~|~\lambda ) = \frac{e^{-\lambda} \lambda ^x}{x!}\quad \mbox{for $x=0,1,2,...$} \]

Question:

Suppose the number of individual plants of a given species we expect to find in a one meter square quadrat follows the Poisson distribution with mean $\lambda= 10$. Find the probability of finding exactly $12$ individuals.

Question:

If there are twelve cars crossing a bridge per minute on average, find the probability of having seventeen or more cars crossing the bridge in a particular minute.

Solutions here

4.3 Aproximation of Binomial as Poisson

Example

Five percent (5%) of Christmas tree light bulbs manufactured by a company are defective. The company’s Quality Control Manager is quite concerned and therefore randomly samples 100 bulbs coming off of the assembly line. Let X denote the number in the sample that are defective. What is the probability that the sample contains at most three defective bulbs?

p = 0.05
k = 3
n = 100
pbinom(k,size=n,prob=p)

## [1] 0.2578387

It can be demonstrated that the Binomial distribution can be approximated with the Poisson probability mass function when $n$ is large. Using the Poisson distribution, the mean $\lambda = np$

lambda <- n*p
sum(dpois(0:3,lambda))

## [1] 0.2650259

It is important to keep in mind that the Poisson approximation to the binomial distribution works well only when $n$ is large and $p$ is small. In general, the approximation works well if $n \geq 20$ and $p\leq0.05$, or if $n\geq 100$ and $p\leq 0.10$.

4.4 Exponential distribution $Exp(\lambda)$

The exponential distribution is the probability distribution that describes the time between events in a Poisson process, i.e. a process in which events occur continuously and independently at a constant average rate. It is a particular case of the gamma distribution. It is the continuous analogue of the geometric distribution, and it has the key property of being memoryless. In addition to being used for the analysis of Poisson processes, it is found in various other contexts.

The probability density function (pdf) of an exponential distribution as \[ f(x;\lambda) = \lambda \exp(-\lambda x) \]

where $\lambda>0$ is the event rate (also known as rate parameter, arrival rate, death rate, failure rate, transition rate). The exponential variable $x \in [0,\infty)$

Cumulative distribution function of the exponential distribution is \[ F(x) = \mbox{Pr}(X\leq x) = \left\{ \begin{array}{lcc} 1- e^{-\lambda x} & & x\geq 0 \\ 0 & & x < 0 \end{array} \right. \]

Mean $\mathbb{E}(X) = 1/\lambda$, and $\mathbb{V}ar(X) = 1/\lambda^2$.

Question:

Suppose that the amount of time one spends in a bank is exponentially distributed with mean 10 minutes, $\lambda=1/10$.

What is the probability that a customer will spend more than 15 minutes in the bank?
What is the probability that a customer will spend more than 15 minutes in the bank given that he is still in the bank after 10 minutes?

Solutions here

4.5 The Normal distribution $\mathcal{N}(\mu,\sigma^2)$

The probability density function of the Normal distribution is:

\[ f(x | \mu,\sigma^2) = \frac{1}{\sqrt{2\sigma^2\pi}} e ^{-\frac{(x-\mu)^2}{2\sigma^2}}, \] where

$\mu$ is the mean of the distribution (also the median and the mode).
$\sigma$ is the standard deviation ($\sigma>0$).
$\sigma^2$ is the variance.

The process to standardized Normal distribution consists in transforming the Normal variable $N(\mu,\sigma)$ to $N(0,1)$, i.e. \[ Z = \frac{X-\mu}{\sigma} \sim N(0,1) \]

Question:

$X$ is a normally distributed variable with mean $\mu = 30$ and standard deviation $\sigma = 4$. Find

$P(x<40)$
$P(x>21)$
$P(30<x<35)$

Question:

Entry to a certain University is determined by a national test. The scores on this test are normally distributed with a mean of 500 and a standard deviation of 100. Tom wants to be admitted to this university and he knows that he must score better than at least 70% of the students who took the test. Tom takes the test and scores 585. Will he be admitted to this university?

Solutions here

4.6 Exercises

A dice is thrown at random. What is the expectation of number on it? (Or) If x denotes the number of points on a dice, find the expectation and the variance of x.

x = 1:6
prob <- 1/6

 E  <- sum(x*rep(prob,length(x)))
 
 E2 <- E^2
 Ex2 <- sum(x^2*rep(prob,length(x)))
Var <- Ex2-E2

If a person gains or loses an amount equal to the number appearing when a balanced die is rolled once, according to whether the number is even or odd, how much money can be expect per game in the long run?

Let $x$ indicate the amount that the person wins

The amount gained by the person would be

- Rs. 1 if the dice shows up ONE
+ Rs. 2 if the dice shows up TWO
- Rs. 3 if the dice shows up THREE
+ Rs. 4 if the dice shows up FOUR
- Rs. 5 if the dice shows up FIVE
+ Rs. 6 if the dice shows up SIX

x = c(1:6)*c(-1,1,-1,1,-1,1)
pr <- rep(1/6,6)
 Ex <- sum(x*pr)
Ex2 <- sum(x^2*pr)
Var <- Ex2-Ex^2
Ex

## [1] 0.5

Var

## [1] 14.91667

5 Statistical Inference

Statistical inference means drawing conclusions based on data. There are many ways of performing inference including statistical modeling, data oriented strategies and explicit use of designs and randomization in analyses. Furthermore, there are broad theories (frequentists, Bayesian, likelihood, design based, …) and numerous complexities (missing data, observed and unobserved confounding, biases) for performing inference. This section presents the fundamentals of inference in a practical approach for getting things done in order to use statistical inference for making informed choices in analyzing data.

5.1 Interval estimation

It is a common requirement to efficiently estimate population parameters based on simple random sample data.

library(MASS)
?survey
head(survey)

##      Sex Wr.Hnd NW.Hnd W.Hnd    Fold Pulse    Clap Exer Smoke Height
## 1 Female   18.5   18.0 Right  R on L    92    Left Some Never 173.00
## 2   Male   19.5   20.5  Left  R on L   104    Left None Regul 177.80
## 3   Male   18.0   13.3 Right  L on R    87 Neither None Occas     NA
## 4   Male   18.8   18.9 Right  R on L    NA Neither None Never 160.00
## 5   Male   20.0   20.0 Right Neither    35   Right Some Never 165.00
## 6 Female   18.0   17.7 Right  L on R    64   Right Some Never 172.72
##        M.I    Age
## 1   Metric 18.250
## 2 Imperial 17.583
## 3     <NA> 16.917
## 4   Metric 20.333
## 5   Metric 23.667
## 6 Imperial 21.000

Point estimate of population mean

For any particular random sample, we can always compute its sample mean. Although most often it is not the actual population mean, it does serve as a good point estimate. For example, in the data set survey, the survey is performed on a sample of the student population. We can compute the sample mean and use it as an estimate of the corresponding population parameter.

Problem: Find a point estimate of mean university student height with the sample data from survey.

mean(survey$Height)

## [1] NA

mean(survey$Height,na.rm=TRUE)

## [1] 172.3809

# or
mean(na.omit(survey$Height))

## [1] 172.3809

After we found a point estimate of the population mean, we would need a way to quantify its accuracy.

5.1.1 Interval Estimate of Population Mean (with known variance)

Let us denote the $100~(1-\alpha/2)$ percentile of the standard normal distribution as $z_{\alpha/2}$. For random sample of sufficiently large size, the end points of the interval estimate at $(1-\alpha/2)$ confidence level is given as follows: \[ \bar{x} \pm \frac{\sigma}{\sqrt{n}} \]

Problem: assume the population standard deviation $\sigma$ of the student height in survey is 9.48. Find the margin of error and interval estimate at 95% confidence level.

x <- na.omit(survey$Height)
n <- length(x)
sigma <- 9.48
sem = sigma/sqrt(n)  # Standard error of the mean
sem

## [1] 0.6557453

Since there are two tails of the normal distribution, the 95% confidence level would imply the 97.5th percentile of the normal distribution at the upper tail. Therefore, $z_{\alpha/2}$ is given by qnorm(.975). We multiply it with the standard error of the mean sem and get the margin of error.

E <- qnorm(0.975)*sem 
E  # margin of error

## [1] 1.285237

xbar <- mean(x,na.rm = TRUE)
xbar + c(-E,E)

## [1] 171.0956 173.6661

Alternatively, we can use z.test

library(TeachingDemos)
z.test(x,sd=sigma)

## 
##  One Sample z-test
## 
## data:  x
## z = 262.88, n = 209.00000, Std. Dev. = 9.48000, Std. Dev. of the
## sample mean = 0.65575, p-value < 2.2e-16
## alternative hypothesis: true mean is not equal to 0
## 95 percent confidence interval:
##  171.0956 173.6661
## sample estimates:
## mean of x 
##  172.3809

Assuming the population standard deviation $\sigma$ being 9.48, the margin of error for the student height survey at 95% confidence level is 1.2852372 centimeters. The confidence interval is between 171.095624 and 173.6660984 centimeters.

5.1.2 Interval Estimate of Population Mean (with unknown variance)

s = sd(x)
SE <- s/sqrt(n)
E = qt(.975, df=n-1)*SE; E

## [1] 1.342878

xbar + c(-E, E)

## [1] 171.0380 173.7237

# or 

t.test(x)

## 
##  One Sample t-test
## 
## data:  x
## t = 253.07, df = 208, p-value < 2.2e-16
## alternative hypothesis: true mean is not equal to 0
## 95 percent confidence interval:
##  171.0380 173.7237
## sample estimates:
## mean of x 
##  172.3809

Without assumption on the population standard deviation, the margin of error for the student height survey at 95% confidence level is 1.3429 centimeters. The confidence interval is between 171.04 and 173.72 centimeters.

5.1.2.1 Sampling Size of Population Mean

The quality of a sample survey can be improved by increasing the sample size. The formula below provide the sample size needed under the requirement of population mean interval estimate at $(1-\alpha)$ confidence level, margin of error $E$, and population variance $\sigma^2$. Here, $z_{\alpha-2}$ is the $100(1 -\alpha/2)$ percentile of the standard normal distribution.

\[ n = \frac{(z_{\alpha/2})^2 \sigma^2}{E^2} \]

Problem:

Assume the population standard deviation $\sigma$ of the student height in survey is 9.48. Find the sample size needed to achieve a 1.2 centimeters margin of error at 95% confidence level.

zstar = qnorm(.975) 
sigma = 9.48 
E = 1.2 
zstar^2 *sigma^2/ E^2

## [1] 239.7454

Based on the assumption of population standard deviation being 9.48, it needs a sample size of 240 to achieve a 1.2 centimeters margin of error at 95% confidence level.

5.1.3 Interval estimates of population proportion

Point estimate of population proportion

Multiple choice questionnaires in a survey are often used to determine the the proportion of a population with certain characteristic. For example, we can estimate the proportion of female students in the university based on the result in the sample data set survey.

Problem: Find a point estimate of the female student proportion from survey.

library(MASS)
gender.response <- na.omit(survey$Sex) # filter out NA's
n <- length(gender.response)
table(gender.response)

## gender.response
## Female   Male 
##    118    118

To find out the number of female students, we compare gender.response

k <- sum(gender.response == "Female") 
# or 
k <- table(gender.response)[1]

pbar <- k/n; pbar

## Female 
##    0.5

The point estimate of the female student proportion in survey is 50%.

In order to compute the confidence interval of the estimated proportion, let us denote the $100(1-\alpha/2)$ percentile of the standard normal distribution as $z_{\alpha/2}$. If the samples size $n$ and population proportion $p$ satisfy the condition that $np \geq 5$ and $n(1-p) \geq 5$, than the end points of the interval estimate at $(1-\alpha)$ confidence level is defined in terms of the sample proportion as follows.

\[ \bar{p} = \pm z_{\alpha/2} \sqrt{\frac{\bar{p}(1-\bar{p})}{n}} \]

We use the prop.test function for direct calculation

prop.test(k,n,conf.level = 0.95)

## 
##  1-sample proportions test without continuity correction
## 
## data:  k out of n, null probability 0.5
## X-squared = 0, df = 1, p-value = 1
## alternative hypothesis: true p is not equal to 0.5
## 95 percent confidence interval:
##  0.4367215 0.5632785
## sample estimates:
##   p 
## 0.5

At 95% confidence level, between 43.6% and 56.3% of the university students are female, and the margin of error is 6.4%.

5.1.3.1 Sampling Size of Population proportion

The quality of a sample survey can be improved by increasing the sample size. The formula below provide the sample size needed under the requirement of population proportion interval estimate at $(1 -\alpha)$ confidence level, margin of error $E$, and planned proportion estimate $p$. Here, $z_{\alpha/2}$ is the $100(1-\alpha/2)$ percentile of the standard normal distribution.

\[ n = \frac{(z_{\alpha/2})^2 p (1-p)}{E^2} \]

zstar = qnorm(.975) 
p = 0.5 
E = 0.05 
zstar^2 * p * (1-p) / E^2

## [1] 384.1459

With a planned proportion estimate of 50% at 95% confidence level, it needs a sample size of 385 to achieve a 5% margin of error for the survey of female student proportion.

5.2 Inference about two populations

It is often necessary to draw conclusion on the difference between two populations by their data samples. In the following tutorials, we discuss how to estimate the difference of means and proportions between two normally distributed data populations.

5.2.1 Population mean between two matches samples

Two data samples are matched if they come from repeated observations of the same subject. Here, we assume that the data populations follow the normal distribution. Using the paired t-test, we can obtain an interval estimate of the difference of the population means.

library(MASS)
data(immer)
?immer
head(immer)

##   Loc Var    Y1    Y2
## 1  UF   M  81.0  80.7
## 2  UF   S 105.4  82.3
## 3  UF   V 119.7  80.4
## 4  UF   T 109.7  87.2
## 5  UF   P  98.3  84.2
## 6   W   M 146.6 100.4

Problem:

Assuming that the data in immer follows the normal distribution, find the 95% confidence interval estimate of the difference between the mean barley yields between years 1931 and 1932.

t.test(immer$Y1, immer$Y2, paired=TRUE)

## 
##  Paired t-test
## 
## data:  immer$Y1 and immer$Y2
## t = 3.324, df = 29, p-value = 0.002413
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##   6.121954 25.704713
## sample estimates:
## mean of the differences 
##                15.91333

Answer:

Between years 1931 and 1932 in the data set immer, the 95% confidence interval of the difference in means of the barley yields is the interval between 6.122 and 25.705.

5.3 Population Mean Between Two Independent Samples

Two data samples are independent if they come from unrelated populations and the samples does not affect each other. Here, we assume that the data populations follow the normal distribution. Using the unpaired t-test, we can obtain an interval estimate of the difference between two population means.

Example:

In the data frame column mpg of the data set mtcars, there are gas mileage data of various 1974 U.S. automobiles. am variable indicates the transmission type of the automobile model (0 = automatic, 1 = manual).

In particular, the gas mileage for manual and automatic transmissions are two independent data populations.

Problem:

Assuming that the data in mtcars follows the normal distribution, find the 95% confidence interval estimate of the difference between the mean gas mileage of manual and automatic transmissions.

t.test(mpg ~ am, data=mtcars)

## 
##  Welch Two Sample t-test
## 
## data:  mpg by am
## t = -3.7671, df = 18.332, p-value = 0.001374
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -11.280194  -3.209684
## sample estimates:
## mean in group 0 mean in group 1 
##        17.14737        24.39231

In mtcars, the mean mileage of automatic transmission is 17.147 mpg and the manual transmission is 24.392 mpg. The 95% confidence interval of the difference in mean gas mileage is between 3.2097 and 11.2802 mpg.

5.4 Comparison of Two Population Proportions

A survey conducted in two distinct populations will produce different results. It is often necessary to compare the survey response proportion between the two populations. Here, we assume that the data populations follow the normal distribution.

Example:

In the built-in data set named quine, children from an Australian town is classified by ethnic background, gender, age, learning status and the number of days absent from school.

library(MASS)
data(quine)
?quine
head(quine)

##   Eth Sex Age Lrn Days
## 1   A   M  F0  SL    2
## 2   A   M  F0  SL   11
## 3   A   M  F0  SL   14
## 4   A   M  F0  AL    5
## 5   A   M  F0  AL    5
## 6   A   M  F0  AL   13

Column Eth indicates whether the student is Aboriginal or Not (“A” or “N”), and the column Sex indicates Male or Female (“M” or “F”).

table(quine$Eth,quine$Sex)

##    
##      F  M
##   A 38 31
##   N 42 35

Problem:

Assuming that the data in quine follows the normal distribution, find the 95% confidence interval estimate of the difference between the female proportion of Aboriginal students and the female proportion of Non-Aboriginal students, each within their own ethnic group.

Solution

We apply the `prop.test function to compute the difference in female proportions. The Yates’s continuity correction is disabled for pedagogical reasons.

prop.test(table(quine$Eth, quine$Sex), correct=FALSE)

## 
##  2-sample test for equality of proportions without continuity
##  correction
## 
## data:  table(quine$Eth, quine$Sex)
## X-squared = 0.0040803, df = 1, p-value = 0.9491
## alternative hypothesis: two.sided
## 95 percent confidence interval:
##  -0.1564218  0.1669620
## sample estimates:
##    prop 1    prop 2 
## 0.5507246 0.5454545

The 95% confidence interval estimate of the difference between the female proportion of Aboriginal students and the female proportion of Non-Aboriginal students is between -15.6% and 16.7%.

5.5 Goodness-of-Fit

Many statistical quantities derived from data samples are found to follow the Chi-squared distribution. Hence we can use it to test whether a population fits a particular theoretical probability distribution.

5.6 Chi-squared Test of Independence

Two random variables $x$ and $y$ are called independent if the probability distribution of one variable is not affected by the presence of another. Assume $f_{ij}$ is the observed frequency count of events belonging to both $i$-th category of x and $j$-th category of $y$. Also assume $e_{ij}$ to be the corresponding expected count if $x$ and $y$ are independent. The null hypothesis of the independence assumption is to be rejected if the p-value of the following Chi-squared test statistics is less than a given significance level $\alpha$.

\[ \chi^2 = \sum_{i,j}\frac{(f_i - e_i)^2}{e_i} \]

Example:

In the built-in data set survey, the Smoke column records the students smoking habit, while the Exer column records their exercise level. The allowed values in Smoke are “Heavy”, “Regul” (regularly), “Occas” (occasionally) and “Never”. As for Exer, they are “Freq” (frequently), “Some” and “None”.

library(MASS)
data(survey)
tbl <- table(survey$Smoke,survey$Exer)
tbl

##        
##         Freq None Some
##   Heavy    7    1    3
##   Never   87   18   84
##   Occas   12    3    4
##   Regul    9    1    7

Problem:

Test the hypothesis whether the students smoking habit is independent of their exercise level at $0.05$ significance level.

chi2test <- chisq.test(tbl) 
chi2test

## 
##  Pearson's Chi-squared test
## 
## data:  tbl
## X-squared = 5.4885, df = 6, p-value = 0.4828

As the p-value is 0.4828422 greater than the 0.05 significance level, we do not reject the null hypothesis that the smoking habit is independent of the exercise level of the students.

The warning message found in the solution above is due to the small cell values in the contingency table. To avoid such warning, we combine the second and third columns of tbl, and save it in a new table named ctbl. Then we apply the chisq.test function against ctbl instead.

ctbl <- cbind(tbl[,"Freq"], tbl[,"None"] + tbl[,"Some"])
ctbl

##       [,1] [,2]
## Heavy    7    4
## Never   87  102
## Occas   12    7
## Regul    9    8

chi2test <- chisq.test(ctbl)
chi2test

## 
##  Pearson's Chi-squared test
## 
## data:  ctbl
## X-squared = 3.2328, df = 3, p-value = 0.3571

The p-value is 0.3571031 also greater than the 0.05 significance level.

5.7 Mann-Whitney U test

Also known as Mann-Whitney-Wilcoxon (MWW), Wilcoxon rank-sum test, or Wilcoxon-Mann-Whitney test is a non-parametric test of the null hypothesis that two samples come from the same sample population against an alternative hypothesis, especially that a particular population tends to have larger values than the other. Is the alternative test to the independent sample t-test. As a non-parametric test, it does not assume any assumptions related to the distribution. Hence, we can decide whether the population distributions are identical without assuming them to follow the normal distribution.

There are, however, some assumptions that are assumed:

The sample drawn from the population is random.
Independence within the samples and mutual independence is assumed.
Ordinal measurement scale is assumed.

Mann-Whitney U test is used for every field, but in frequently used in psychology, medical/nursing and business. For example, in psychology, it is used to compare attitude or behavior, etc. In medicine, it is used to know the effect of two medicines and whether they are equal or not. It is also used to know whether or not a particular medicine cures the ailment or not. In business, it can be used to know the preferences of different people and it can be used to see if that changes depending on location.

Example Consider the mtcars data set. We are aim to find out whether automatic transmission cars are less fuel efficient compared to manual transmission cars. Since 1950, most cars sold in North America have been available with an automatic transmission.

The mtcars dataset is from the 1974 “Motor Trend US” magazine, consisting of fuel consumption measurement (mpg) and 10 different aspects of automoblie design and performance for 32 automobiles (1973-74 models).

We first plot the data (mpg vs transmission)

either with a boxplot

boxplot(mpg~am,data=mtcars,names=c("automatic","manual"),ylab="mpg",xlab="Transmission")

or with a bar chart (with error bars)

We run some statistical tests to compare mpg between automatic- and manual-transmitted cars.

The first one is a t-test, assuming that the mileage data have a normal distribution. The test result clearly shows that the manual transmission cars are more gas-efficient than automatic transmission cars (miles per gallon: 24.39 versus 17.15).

t.test(mpg ~ factor(am), data = mtcars)

## 
##  Welch Two Sample t-test
## 
## data:  mpg by factor(am)
## t = -3.7671, df = 18.332, p-value = 0.001374
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -11.280194  -3.209684
## sample estimates:
## mean in group 0 mean in group 1 
##        17.14737        24.39231

However, the normality assumption could be quite strong, given the fact that we do not know the true underlying distributions of mpg data. Moreover, the number of data points are not large enough to apply the central limit theorem. Therefore, a more conservative test would be the Wilcoxon test, with the null hypothesis that the gas mileage data of manual and automatic transmissions are from identical populations.

wilcox.test(mpg ~ am, data = mtcars)

## Warning in wilcox.test.default(x = c(21.4, 18.7, 18.1, 14.3, 24.4, 22.8, :
## cannot compute exact p-value with ties

## 
##  Wilcoxon rank sum test with continuity correction
## 
## data:  mpg by am
## W = 42, p-value = 0.001871
## alternative hypothesis: true location shift is not equal to 0

A non-parametric, Wilcoxon test also rejects the null hypothesis that the mileage data of the manual and automatic transmissions are from the same population (indicating a difference).

6 Linear models in `R`

6.1 Simple linear regression

Regression is a statistical method used to predict the value of a response variable based on the values of a set of explanatory variables.
One very general form for the model would be

\[ y = f(x_1,x_2,...,x_p) + \epsilon, \]

where $f$ is some unknown function and $\epsilon$ is the error in this representation. Since we usually don’t have enough data to try to estimate $f$ directly (inverse problem), we usually have to assume that it has some restricted form.

Any statistical model attempts to approximate the response variable or dependent variable $y$ as a mathematical function of the explanatory variables or regressors $X$ (also called covariates or independent variables).
The simplest and most common form is the linear model (LM)

\[ y = \beta_0 + \beta_1 x_1 + \beta_2 x_z + \epsilon, \] where $\beta_i$ $i=0,1,2$ are unknown parameters. $\beta_0$ is called the intercept term. Hence, the problem is reduced to the estimation of four values rather than the complicated infinite dimensional $f$.

A simple linear model with a single exploratory variable is defined as: \[ \hat{y} = \beta_0 + \beta_1 x \]

where $\hat y$ is the fitted values for $\beta_0$ (intercept) and $\beta_1$ (slope). Then for a given $x_i$ we obtain a $\hat{y}_i$ that approximates $y_i$

Let us create a toy example (with $p=1$):

set.seed(1)
n <- 50 

x <- seq(1,n)
 beta0 <- 15
 beta1 <- 0.5

sigma <- 3 # standar deviation of the errors
eps <- rnorm(n,mean=0,sd=3) # generate gaussian random errors

# Generate random data
 y <- beta0 + beta1*x  +  eps

Plot the data

plot(x,y,ylim = c(8,45), cex=1.3, xlab = "x", ylab="y",pch=19)

A mathematical procedure for finding the best-fitting curve to a given set of points by minimizing the sum of the squares of the residuals of the points from the fitted line. Illustration of the least squares fit

For example, in the data set faithful, it contains sample data of two random variables named waiting and eruptions. The waiting variable denotes the waiting time until the next eruptions, and eruptions denotes the duration.

plot(eruptions~waiting,data=faithful)

Its linear regression model can be expressed as:

\[ Eruptions = \beta_0 + \beta_1*Waiting + \epsilon \]

If we choose the parameters $\beta_0$ and $\beta_1$ in the simple linear regression model so as to minimize the sum of squares of the error term $\epsilon$. Suppose for the data set faithful, we aim to estimate the next eruption duration if the waiting time since the last eruption has been 80 minutes.

We apply the lm function to a formula that describes the variable eruptions by the variable waiting, and save the linear regression model in a new variable eruption.lm.

data("faithful")
eruption.lm <- lm(eruptions~waiting,data=faithful)

Then we extract the parameters of the estimated regression equation with the coefficients function.

coeffs <- coefficients(eruption.lm); coeffs

## (Intercept)     waiting 
## -1.87401599  0.07562795

We now fit the eruption duration using the estimated regression equation.

waiting = 80           # the waiting time 
duration = coeffs[1] + coeffs[2]*waiting 
duration

## (Intercept) 
##     4.17622

Based on the simple linear regression model, if the waiting time since the last eruption has been 80 minutes, we expect the next one to last 4.1762198 minutes.

We wrap the waiting parameter value inside a new data frame named newdata.

newdata = data.frame(waiting=80) # wrap the parameter

Then we apply the predict function to eruption.lm along with newdata.

predict(eruption.lm, newdata)    # apply predict

##       1 
## 4.17622

We can directly calculate quantities of interest, i.e. the ordinary least squares solution consists of:

\[ \min_{\beta_0,\beta_1} = \sum_{i=1}^{n} (y_i - \hat{y}_i)^2 \]

Then $\hat{\beta}_1 = \frac{\sum_{i=1}^{n}x_iy_i}{\sum_{i=1}^n x_i^2}$ and $\hat{\beta}_0 = \bar{y} - \hat{\beta}_1\bar{x}$

In matrix form, with $X=[1:x_1:...:x_p]$

\[ \hat{\beta} = (X^\prime X)^{-1} X^\prime \]

where $\hat{\beta} = (\hat{\beta}_0,\hat{\beta}_1)$

6.2 Defining models in `R`

To complete a linear regression using R it is first necessary to understand the syntax for defining models.

Syntax	Model	Comments
`y ~ x`	$y = \beta_0+\beta_1x$	Straight-line with an implicit intercept
`y ~ -1 + x`	$y = \beta_1x$	Straight-line with no intercept; that is, a fit forced through (0,0)
`y ~ x + I(x^2)`	$y = \beta_0+\beta_1x+\beta_2x^2$	Polynomial model; `I()` allows for mathematical symbols
`y ~ x + z`	$y = \beta_0+\beta_1x+\beta_2z$	Multiple regression model
`y ~ x:z`	$y = \beta_0+\beta_1xz$	Model with interaction between $x$ and $z$
`y ~ x*z`	$y = \beta_0+\beta_1x+\beta_2z+\beta_3xz$	Equivalent to `y~x+z+x:z`

In R function model.matrix helps us to create the $X$ matrix.

x <- model.matrix(~waiting, data = faithful)
y <- faithful$eruptions
xtxi <- solve(t(x) %*% x)
betas <- xtxi %*% t(x) %*% y
betas

##                    [,1]
## (Intercept) -1.87401599
## waiting      0.07562795

solve(crossprod(x, x), crossprod(x, y))

##                    [,1]
## (Intercept) -1.87401599
## waiting      0.07562795

Of course, it is not necessary here because the lm() function does the job but it is very useful when the statistic you want is not part of the pre-packaged functions.

It is possible to get $(X^\prime X)^{-1}$ as

summary(eruption.lm)$cov.unscaled

##              (Intercept)       waiting
## (Intercept)  0.104029479 -1.415475e-03
## waiting     -0.001415475  1.996521e-05

The names() command is the way to see the components of an R object

names(eruption.lm)

##  [1] "coefficients"  "residuals"     "effects"       "rank"         
##  [5] "fitted.values" "assign"        "qr"            "df.residual"  
##  [9] "xlevels"       "call"          "terms"         "model"

To access for example

Fitted (or predictd) values ($\hat{y}$):

eruption.lm$fitted.values

Residuals ($y-\hat{y}$)

eruption.lm$residuals

We can estimate $\sigma$ as $\sigma = \frac{(y_i-\hat{y}_i)^2}{n-p}$ in R

eruption.lm.sum <- summary(eruption.lm)
names(eruption.lm.sum)

##  [1] "call"          "terms"         "residuals"     "coefficients" 
##  [5] "aliased"       "sigma"         "df"            "r.squared"    
##  [9] "adj.r.squared" "fstatistic"    "cov.unscaled"

sqrt(deviance(eruption.lm)/df.residual(eruption.lm))

## [1] 0.4965129

# is obtained directly as
eruption.lm.sum$sigma

## [1] 0.4965129

We may also obtain the standard errors for the coefficients. Also diag() returns the diagonal of a matrix:

xtxi

##              (Intercept)       waiting
## (Intercept)  0.104029479 -1.415475e-03
## waiting     -0.001415475  1.996521e-05

sqrt(diag(xtxi)) * eruption.lm.sum$sigma

## (Intercept)     waiting 
## 0.160143302 0.002218541

eruption.lm.sum$coef[, 2]

## (Intercept)     waiting 
## 0.160143302 0.002218541

6.3 Coefficient of Determination

The coefficient of determination of a linear regression model is the quotient of the variances of the fitted values and observed values of the dependent variable. If we denote $y_i$ as the observed values of the dependent variable, $\bar{y}$ as its mean, and $\bar{y}_i$ as the fitted value, then the coefficient of determination is: \[ R^2 = \frac{\sum (\hat{y}_i-\bar{y})^2}{(y_i - \bar{y})^2} \]

summary(eruption.lm)$r.squared

## [1] 0.8114608

1-sum(eruption.lm$res^2)/sum((y-mean(y))^2)

## [1] 0.8114608

More options:

fitted.values() or fitted() Fitted values of the model
predict(): to predict $\hat{y}_*$ for new values of $x_*$
confint(): confidence intervals for model parameters
resid(): residuals of the model
anova(): ANOVA table for residuals
deviance(): deviance of the fitted model, in the LM case is $\sum_i^{n}(\hat{y}_i - y_i)^2$

See Faraway’s (2002) book (Chapters 1-7)

6.4 Significance Test for Linear Regression

Assume that the error term ?? in the linear regression model is independent of x, and is normally distributed, with zero mean and constant variance. We can decide whether there is any significant relationship between x and y by testing the null hypothesis that $\beta_1 = 0$.

we print out the F-statistics of the significance test with the summary function.

summary(eruption.lm)

## 
## Call:
## lm(formula = eruptions ~ waiting, data = faithful)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.29917 -0.37689  0.03508  0.34909  1.19329 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -1.874016   0.160143  -11.70   <2e-16 ***
## waiting      0.075628   0.002219   34.09   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.4965 on 270 degrees of freedom
## Multiple R-squared:  0.8115, Adjusted R-squared:  0.8108 
## F-statistic:  1162 on 1 and 270 DF,  p-value: < 2.2e-16

6.5 Confidence Interval for Linear Regression

Assume that the error term $\epsilon$ in the linear regression model is independent of $x$, and is normally distributed, with zero mean and constant variance. For a given value of $x$, the interval estimate for the mean of the dependent variable, $\bar{y}$ , is called the confidence interval.

A 95% confidence interval of the mean eruption duration for the waiting time of 80 minutes is given by

predict(eruption.lm, newdata, interval="confidence")

##       fit      lwr      upr
## 1 4.17622 4.104848 4.247592

The 95% confidence interval of the mean eruption duration for the waiting time of 80 minutes is between 4.1048 and 4.2476 minutes.

6.6 Prediction Interval for Linear Regression

For a given value of $x$, the interval estimate of the dependent variable $y$ is called the prediction interval.

predict(eruption.lm, newdata, interval="predict")

##       fit      lwr      upr
## 1 4.17622 3.196089 5.156351

The 95% prediction interval of the eruption duration for the waiting time of 80 minutes is between 3.1961 and 5.1564 minutes.

6.7 Residual Plot

The residual data of the simple linear regression model is the difference between the observed data of the dependent variable $y$ and the fitted values $\hat{y}$.

\[ Residual = y -\hat{y} \]

eruption.res = resid(eruption.lm)

plot(faithful$waiting,
    eruption.res, 
    ylab="Residuals", xlab="Waiting Time", 
    main="Old Faithful Eruptions") 
abline(0, 0)

6.8 Standardized Residual

The standardized residual is the residual divided by its standard deviation.

\[ \mbox{Standardized residual}_i = \frac{Residual_i}{SD.of.Residual_i} \]

eruption.stdres <- rstandard(eruption.lm)

plot(faithful$waiting, eruption.stdres, 
      ylab="Standardized Residuals", 
      xlab="Waiting Time", 
      main="Old Faithful Eruptions") 
abline(0, 0)

As the p-value is much less than $0.05$, we reject the null hypothesis that $\beta_1 = 0$. Hence there is a significant relationship between the variables in the linear regression model of the data set faithful.

6.9 Normal Probability Plot of Residuals

The normal probability plot is a graphical tool for comparing a data set with the normal distribution. We can use it with the standardized residual of the linear regression model and see if the error term $\epsilon$ is actually normally distributed.

eruption.lm = lm(eruptions ~ waiting, data=faithful) 
eruption.stdres = rstandard(eruption.lm)

We now create the normal probability plot with the qqnorm function, and add the qqline for further comparison.

 qqnorm(eruption.stdres, 
     ylab="Standardized Residuals", 
     xlab="Normal Scores", 
     main="Old Faithful Eruptions") 
 qqline(eruption.stdres)

6.10 Multiple linear regression

A multiple linear regression (MLR) model that describes a dependent variable $y$ by independent variables $x_1, x_2, ..., x_p$ $(p > 1)$ is expressed by the equation: \[ y = \beta_0 + \sum_{k}^{p} \beta_k + \epsilon \] where the numbers $\beta_0$ and $\beta_k$ ($k = 1, 2, ..., p$) are the parameters, and $\epsilon$ is the error term.

Example:

Consider the data stackloss from observations of a chemical plant operation, if we assign stackloss as the dependent variable, and assign Air.Flow (cooling air flow), Water.Temp (inlet water temperature) and Acid.Conc. (acid concentration) as independent variables, the multiple linear regression model is:

\[ stack.loss = \beta_0 + \beta_1 * Air.Flow + \beta_2 * Water.Temp + \beta_3 * Acid.Conc + \epsilon \]

data("stackloss")
?stackloss
head(stackloss)

##   Air.Flow Water.Temp Acid.Conc. stack.loss
## 1       80         27         89         42
## 2       80         27         88         37
## 3       75         25         90         37
## 4       62         24         87         28
## 5       62         22         87         18
## 6       62         23         87         18

Fit the multiple linear regression in R

stackloss.lm = lm(stack.loss ~ Air.Flow + Water.Temp + Acid.Conc., data=stackloss) 
stackloss.lm

## 
## Call:
## lm(formula = stack.loss ~ Air.Flow + Water.Temp + Acid.Conc., 
##     data = stackloss)
## 
## Coefficients:
## (Intercept)     Air.Flow   Water.Temp   Acid.Conc.  
##    -39.9197       0.7156       1.2953      -0.1521

summary(stackloss.lm)

## 
## Call:
## lm(formula = stack.loss ~ Air.Flow + Water.Temp + Acid.Conc., 
##     data = stackloss)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -7.2377 -1.7117 -0.4551  2.3614  5.6978 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -39.9197    11.8960  -3.356  0.00375 ** 
## Air.Flow      0.7156     0.1349   5.307  5.8e-05 ***
## Water.Temp    1.2953     0.3680   3.520  0.00263 ** 
## Acid.Conc.   -0.1521     0.1563  -0.973  0.34405    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 3.243 on 17 degrees of freedom
## Multiple R-squared:  0.9136, Adjusted R-squared:  0.8983 
## F-statistic:  59.9 on 3 and 17 DF,  p-value: 3.016e-09

Function termplot plots regression terms against their predictors:

?termplot
par(mfrow=c(2,2))
termplot(stackloss.lm, partial.resid = TRUE, se=TRUE,col.se = "blue")

What is the stack loss if the air flow is 72, water temperature is 20 and acid concentration is 85?

Create a new data frame:

newdata <- data.frame(Air.Flow=72,Water.Temp=20,Acid.Conc.=85)

Use predict

predict(stackloss.lm, newdata)

##        1 
## 24.58173

Based on the multiple linear regression model and the given parameters, the predicted stack loss is 24.5817284.

To obtain the multiple coefficient of determination

summary(stackloss.lm)$r.squared

## [1] 0.9135769

6.10.1 Adjusted coefficient of determination

The adjusted coefficient of determination of a multiple linear regression model is defined in terms of the coefficient of determination as follows, where $n$ is the number of observations in the data set, and $p$ is the number of independent variables.

\[ R^2_{adj} = 1-(1-R^2)\frac{n-1}{n-p-1} \]

summary(stackloss.lm)$adj.r.squared

## [1] 0.8983258

6.10.2 Significant tests and confidence/prediction intervals

summary(stackloss.lm)

## 
## Call:
## lm(formula = stack.loss ~ Air.Flow + Water.Temp + Acid.Conc., 
##     data = stackloss)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -7.2377 -1.7117 -0.4551  2.3614  5.6978 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -39.9197    11.8960  -3.356  0.00375 ** 
## Air.Flow      0.7156     0.1349   5.307  5.8e-05 ***
## Water.Temp    1.2953     0.3680   3.520  0.00263 ** 
## Acid.Conc.   -0.1521     0.1563  -0.973  0.34405    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 3.243 on 17 degrees of freedom
## Multiple R-squared:  0.9136, Adjusted R-squared:  0.8983 
## F-statistic:  59.9 on 3 and 17 DF,  p-value: 3.016e-09

As the p-values of Air.Flow and Water.Temp are less than 0.05, they are both statistically significant in the multiple linear regression model of stackloss.

95% Confidence intervals of the stack loss if the air flow is 72, water temperature is 20 and acid concentration is 85.

predict(stackloss.lm, newdata, interval="confidence")

##        fit      lwr    upr
## 1 24.58173 20.21846 28.945

95% Prediction intervals are

predict(stackloss.lm, newdata, interval="prediction")

##        fit     lwr      upr
## 1 24.58173 16.4661 32.69736

6.11 Linear regression with factor variables

Let us consider the mtcars we analyzed previously. In Section.

data(mtcars)
t.test(mpg ~ am, data=mtcars)

## 
##  Welch Two Sample t-test
## 
## data:  mpg by am
## t = -3.7671, df = 18.332, p-value = 0.001374
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -11.280194  -3.209684
## sample estimates:
## mean in group 0 mean in group 1 
##        17.14737        24.39231

The results from the statistical tests focus on mpg and am only, without controlling for influences from other variables.

The benefit of regressional analysis serve the purpose. If we apply a multivariate regression to control for certain available design and performance variables, the marginal impact of automatic- or manual- transmission cars does not turn out to be significant. The confounding variables include displacement (disp), rear axle ratio (drat) and car weight(wt). Take car weight for example. The regression suggests that, holding other variables constant (ceteris paribus), maual tramsmitted cars consume on avarage -0.024 more gallons of gas per mile, and the results are no longer statistically significant. Similar analysis work similarily for the other two variables: drat and wt.

fit0 <- lm(mpg ~ factor(am), data = mtcars)
summary(fit0)

## 
## Call:
## lm(formula = mpg ~ factor(am), data = mtcars)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -9.3923 -3.0923 -0.2974  3.2439  9.5077 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   17.147      1.125  15.247 1.13e-15 ***
## factor(am)1    7.245      1.764   4.106 0.000285 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.902 on 30 degrees of freedom
## Multiple R-squared:  0.3598, Adjusted R-squared:  0.3385 
## F-statistic: 16.86 on 1 and 30 DF,  p-value: 0.000285

If we apply a multivariate regression to control for certain available design and performance variables, the marginal impact of automatic- or manual- transmission cars does not turn out to be significant. The confounding variables include displacement (disp), rear axle ratio (drat) and car weight(wt). Take car weight for example:

fit1 <- lm(mpg ~ factor(am) + wt, data = mtcars)
summary(fit1)

## 
## Call:
## lm(formula = mpg ~ factor(am) + wt, data = mtcars)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -4.5295 -2.3619 -0.1317  1.4025  6.8782 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 37.32155    3.05464  12.218 5.84e-13 ***
## factor(am)1 -0.02362    1.54565  -0.015    0.988    
## wt          -5.35281    0.78824  -6.791 1.87e-07 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 3.098 on 29 degrees of freedom
## Multiple R-squared:  0.7528, Adjusted R-squared:  0.7358 
## F-statistic: 44.17 on 2 and 29 DF,  p-value: 1.579e-09

The regression suggests that, holding other variables constant, manual tramsmitted cars consume on average -0.024 more gallons of gas per mile, and the results are no longer statistically significant. Similar analysis work similarily for the other two variables: drat and wt.

fit2 <- lm(mpg ~ factor(am) + drat, data = mtcars)
summary(fit2)

## 
## Call:
## lm(formula = mpg ~ factor(am) + drat, data = mtcars)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -9.5802 -2.5206 -0.5153  2.4419  8.5198 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)  
## (Intercept)   -1.950      7.073  -0.276   0.7848  
## factor(am)1    2.807      2.282   1.230   0.2286  
## drat           5.811      2.130   2.728   0.0107 *
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.448 on 29 degrees of freedom
## Multiple R-squared:  0.4906, Adjusted R-squared:  0.4554 
## F-statistic: 13.96 on 2 and 29 DF,  p-value: 5.659e-05

fit3 <- lm(mpg ~ factor(am) + disp, data = mtcars)
summary(fit3)

## 
## Call:
## lm(formula = mpg ~ factor(am) + disp, data = mtcars)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -4.6382 -2.4751 -0.5631  2.2333  6.8386 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 27.848081   1.834071  15.184 2.45e-15 ***
## factor(am)1  1.833458   1.436100   1.277    0.212    
## disp        -0.036851   0.005782  -6.373 5.75e-07 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 3.218 on 29 degrees of freedom
## Multiple R-squared:  0.7333, Adjusted R-squared:  0.7149 
## F-statistic: 39.87 on 2 and 29 DF,  p-value: 4.749e-09

We can use termplot

termplot(fit0,partial.resid = TRUE,se=TRUE)

par(mfrow=c(1,2))
termplot(fit1,partial.resid = TRUE,se=TRUE)

termplot(fit2,partial.resid = TRUE,se=TRUE)

6.12 Inference

Hypothesis test to compare models: the Likelihood ratio test

A likelihood ratio test is a statistical test used to compare the goodness of fit of two models, one of which (the null model) is a special case of the other (the alternative model). The test is based on the likelihood ratio, which expresses how many times more likely the data are under one model than the other.

Comparing two models fit to the same data can be set up as a hypothesis testing problem. Let $M_0$ and $M_1$ denote the models. Consider as the null hypothesis “$M_1$ is not a significant improvement on $M_0$”, and the alternative the negation. This hypothesis can often be formulated so that a statistic can be generated from the two models.

Normally, the models are nested in that the variables in $M_0$ are a subset of those in $M_1$. The statistic often involves the $RSS$ (residual sum of squares) values for both models, adjusted by the number of parameters used. In linear regression this becomes an anova test (comparing variances).

More robust is a likelihood ratio test for nested models. When models are sufficiently specific to define a probability distribution for $y$, the model will report the log-likelihood, $\hat{L}$. Under some mild assumptions, $-2(\hat{L}_0 - \hat{L}_1)$ follows a Chi-squared distribution with degrees of freedom = difference in number of parameters on the two models.

The utility of a single model $M_1$ is often assessed by comparing it with the null model, that reflects no dependence of $y$ on the explanatory variables. The model formula for the null model is y~1, i.e. we use a constant to approximate y (e.g.: the mean of y). The likelihood ratio test is implemented in the function anova:

M0 <- lm(mpg ~ 1, data = mtcars)
M1 <- lm(mpg ~ factor(am), data = mtcars)
summary(M0)

## 
## Call:
## lm(formula = mpg ~ 1, data = mtcars)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -9.6906 -4.6656 -0.8906  2.7094 13.8094 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   20.091      1.065   18.86   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 6.027 on 31 degrees of freedom

summary(M1)

## 
## Call:
## lm(formula = mpg ~ factor(am), data = mtcars)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -9.3923 -3.0923 -0.2974  3.2439  9.5077 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   17.147      1.125  15.247 1.13e-15 ***
## factor(am)1    7.245      1.764   4.106 0.000285 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.902 on 30 degrees of freedom
## Multiple R-squared:  0.3598, Adjusted R-squared:  0.3385 
## F-statistic: 16.86 on 1 and 30 DF,  p-value: 0.000285

anova(M0,M1)

## Analysis of Variance Table
## 
## Model 1: mpg ~ 1
## Model 2: mpg ~ factor(am)
##   Res.Df    RSS Df Sum of Sq     F   Pr(>F)    
## 1     31 1126.0                                
## 2     30  720.9  1    405.15 16.86 0.000285 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

The likelihood ratio test can also test the significance of predictors. Hence, we can compare the model fit0 (where am is significant) with fit1, fit2 or fit3, i.e.:

anova(fit0, fit1)

## Analysis of Variance Table
## 
## Model 1: mpg ~ factor(am)
## Model 2: mpg ~ factor(am) + wt
##   Res.Df    RSS Df Sum of Sq      F    Pr(>F)    
## 1     30 720.90                                  
## 2     29 278.32  1    442.58 46.115 1.867e-07 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

anova(fit0, fit2)

## Analysis of Variance Table
## 
## Model 1: mpg ~ factor(am)
## Model 2: mpg ~ factor(am) + drat
##   Res.Df    RSS Df Sum of Sq      F Pr(>F)  
## 1     30 720.90                             
## 2     29 573.64  1    147.26 7.4444 0.0107 *
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

anova(fit0, fit3)

## Analysis of Variance Table
## 
## Model 1: mpg ~ factor(am)
## Model 2: mpg ~ factor(am) + disp
##   Res.Df    RSS Df Sum of Sq      F    Pr(>F)    
## 1     30 720.90                                  
## 2     29 300.28  1    420.62 40.621 5.748e-07 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

However, the likelihood ratio tests suggest that it is important to consider these dimensions (i.e., displacement, rear axle ratio and weight) since these variables increase model fit.

6.13 ANALYSIS-OF-VARIANCE (ANOVA)

In an experiment study, various treatments are applied to test subjects and the response data is gathered for analysis. A critical tool for carrying out the analysis is the Analysis of Variance (ANOVA). It enables a researcher to differentiate treatment results based on easily computed statistical quantities from the treatment outcome.

The statistical process is derived from estimates of the population variances via two separate approaches. The first approach is based on the variance of the sample means, and the second one is based on the mean of the sample variances. Under the ANOVA assumptions as stated below, the ratio of the two statistical estimates follows the F distribution. Hence we can test the null hypothesis on the equality of various response data from different treatments via estimates of critical regions.

The treatment responses are independent of each other.
The response data follow the normal distribution.
The variances of the response data are identical.

In the following tutorials, we demonstrate how to perform ANOVA on a few basic experimental designs.

# Two-way Interaction Plot
data(mtcars)
attach(mtcars)

## The following objects are masked from mtcars (pos = 11):
## 
##     am, carb, cyl, disp, drat, gear, hp, mpg, qsec, vs, wt

gear <- factor(gear)
cyl <- factor(cyl)
interaction.plot(cyl, gear, mpg, type="b", col=c(1:3),
   leg.bty="o", leg.bg="beige", lwd=2, pch=c(18,24,22),
   xlab="Number of Cylinders",
   ylab="Mean Miles Per Gallon",
   main="Interaction Plot")

# Plot Means with Error Bars
library(gplots)
attach(mtcars)

## The following objects are masked _by_ .GlobalEnv:
## 
##     cyl, gear

## The following objects are masked from mtcars (pos = 3):
## 
##     am, carb, cyl, disp, drat, gear, hp, mpg, qsec, vs, wt

## The following objects are masked from mtcars (pos = 12):
## 
##     am, carb, cyl, disp, drat, gear, hp, mpg, qsec, vs, wt

cyl <- factor(cyl)
plotmeans(mpg~cyl,xlab="Number of Cylinders",
  ylab="Miles Per Gallon", main="Mean Plot\nwith 95% CI")

7 Logistic regression

A logistic regression is typically used when there is one dichotomous outcome variable (such as winning or losing), and a continuous predictor variable which is related to the probability or odds of the outcome variable. It can also be used with categorical predictors, and with multiple predictors.

7.1 ESR and Plasma Proteins

The erythrocyte sedimentation rate (ESR) is the rate at which red blood cells (erythrocytes) settle out of suspension in blood plasma, when measured under standard conditions. If the ESR increases when the level of certain proteins in the blood plasma rise in association with conditions such as rheumatic diseases, chronic infections and malignant diseases, its determination might be useful in screening blood samples taken from people suspected of suffering from one of the conditions mentioned. The absolute value of the ESR is not of great importance; rather, less than 20mm/hr indicates a “healthy” individual. To assess whether the ESR is a useful diagnostic tool, the question of interest is whether there is any association between the probability of an ESR reading greater than 20mm/hr and the levels of the two plasma proteins. If there is not then the determination of ESR would not be useful for diagnostic purposes. A data frame with 32 observations on the following 3 variables.

fibrinogen the fibrinogen level in the blood.
globulin the globulin level in the blood.
ESR the erythrocyte sedimentation rate, either less or greater 20 mm / hour.

data("plasma", package = "HSAUR")
head(plasma)

##   fibrinogen globulin      ESR
## 1       2.52       38 ESR < 20
## 2       2.56       31 ESR < 20
## 3       2.19       33 ESR < 20
## 4       2.18       31 ESR < 20
## 5       3.41       37 ESR < 20
## 6       2.46       36 ESR < 20

layout(matrix(1:2, ncol = 2))
boxplot(fibrinogen ~ ESR, data = plasma, varwidth = TRUE, main="Fibrinogen level in the blood")
boxplot(globulin ~ ESR, data = plasma, varwidth = TRUE, main="Globulin level in the blood")

The question of interest is whether there is any association between the probability of an ESR reading greater than 20mm/hr and the levels of the two plasma proteins. If there is not then the determination of ESR would not be useful for diagnostic purposes.

Since the response variable is binary, a multiple regression model is not suitable for a regression analysis.

We may write \[ \mathbb{P}\mbox{r}(y_i=1)=\pi_i \qquad \mathbb{P}\mbox{r}(y_i=0)=1-\pi_i \]

Normally, we will have a set of covariates $X=(x_1,..., x_p)$ associated with each individual, and our goal will be to investigate the relationship between the response probability $\pi=\pi(X)$ and the explanatory variables.

So instead of modelling the expected value of the response directly as a linear function of explanatory variables, a suitable transformation is modelled. In this case the most suitable transformation is the logistic or logit function of $\pi$ leading to the model

\[ \mbox{logit}(\pi) = \mbox{logit}\left(\frac{\pi}{1-\pi}\right) = \beta_0 + \beta_1x_1 + ... + \beta_p x_p \]

The logit of a probability is simply the log of the odds of the response taking the value one or logit transformation, of p: $logit(p) = \log(p/1-p)$. Logit is sometimes called “log odds.” Because of the properties of odds given in the list above, the logit has these properties:

If odds(y=1s) = 1, then logit(p) = 0.
If odds(y=1) < 1, then logit(p) < 0.
If odds(y=1) > 1, then logit(p) > 0.

The logit transform fails if p = 0`.

When the response is a binary (dichotomous) variable, and x is numeric, logistic regression fits a logistic curve to the relationship between $x$ and $y$. Hence, logistic regression is linear regression on the logit transform of y, where y is the proportion (or probability) of success at each value of x. However, you should avoid the temptation to do a traditional least-squares regression at this point, as neither the normality nor the homoscedasticity assumption will be met.

x <- seq(-6,6,0.01)
logistic <- exp(x)/(1+exp(x))
plot(x,logistic,t='l',main="Logistic curve",ylab="")
abline(h=c(0,0.5,1),v=0,col="grey")
points(0,0.5,pch=19,col=2)

Logistic regression model in R can be fitted using the function glm. First, we start with a model that includes only a single predictor fibrinogen

plasma_glm_1 <- glm(ESR ~ fibrinogen, data = plasma,family = binomial())
summary(plasma_glm_1)

## 
## Call:
## glm(formula = ESR ~ fibrinogen, family = binomial(), data = plasma)
## 
## Deviance Residuals: 
##     Min       1Q   Median       3Q      Max  
## -0.9298  -0.5399  -0.4382  -0.3356   2.4794  
## 
## Coefficients:
##             Estimate Std. Error z value Pr(>|z|)  
## (Intercept)  -6.8451     2.7703  -2.471   0.0135 *
## fibrinogen    1.8271     0.9009   2.028   0.0425 *
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 30.885  on 31  degrees of freedom
## Residual deviance: 24.840  on 30  degrees of freedom
## AIC: 28.84
## 
## Number of Fisher Scoring iterations: 5

We see that the regression coefficient for fibrinogen is significant at the $5\%$ level. An increase of one unit in this variable increases the log-odds in favour of an ESR value greater than 20 by an estimated $1.83$ with $95\%$ confidence interval.

confint(plasma_glm_1,parm="fibrinogen")

## Waiting for profiling to be done...

##     2.5 %    97.5 % 
## 0.3387619 3.9984921

These values are more helpful if converted to the corresponding values for the odds themselves by exponentiating the estimate

exp(coef(plasma_glm_1)["fibrinogen"])

## fibrinogen 
##   6.215715

and the confidence interval

exp(confint(plasma_glm_1, parm = "fibrinogen"))

## Waiting for profiling to be done...

##     2.5 %    97.5 % 
##  1.403209 54.515884

The confidence interval is very wide because there are few observations overall and very few where the ESR value is greater than 20. Nevertheless it seems likely that increased values of fibrinogen lead to a greater probability of an ESR value greater than 20. We can now fit a logistic regression model that includes both explanatory variables using the code

plasma_glm_2 <- glm(ESR ~ fibrinogen + globulin, data = plasma,family = binomial())
summary(plasma_glm_2)

## 
## Call:
## glm(formula = ESR ~ fibrinogen + globulin, family = binomial(), 
##     data = plasma)
## 
## Deviance Residuals: 
##     Min       1Q   Median       3Q      Max  
## -0.9683  -0.6122  -0.3458  -0.2116   2.2636  
## 
## Coefficients:
##             Estimate Std. Error z value Pr(>|z|)  
## (Intercept) -12.7921     5.7963  -2.207   0.0273 *
## fibrinogen    1.9104     0.9710   1.967   0.0491 *
## globulin      0.1558     0.1195   1.303   0.1925  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 30.885  on 31  degrees of freedom
## Residual deviance: 22.971  on 29  degrees of freedom
## AIC: 28.971
## 
## Number of Fisher Scoring iterations: 5

The coefficient for gamma globulin is not significantly different from zero.

Both nested models can be compared using a likelihood ratio test with anova function

anova(plasma_glm_1, plasma_glm_2, test = "Chisq")

## Analysis of Deviance Table
## 
## Model 1: ESR ~ fibrinogen
## Model 2: ESR ~ fibrinogen + globulin
##   Resid. Df Resid. Dev Df Deviance Pr(>Chi)
## 1        30     24.840                     
## 2        29     22.971  1   1.8692   0.1716

So we conclude that gamma globulin is not associated with ESR level.

The plot clearly shows the increasing probability of an ESR value above 20 (larger circles) as the values of fibrinogen, and to a lesser extent, gamma globulin, increase.

prob <- predict(plasma_glm_2,type="response")
plot(globulin ~ fibrinogen, data = plasma, xlim = c(2, 6),ylim = c(25, 55), pch = ".")
symbols(plasma$fibrinogen, plasma$globulin, circles = prob,add = TRUE)

8 Advanced graphics in `R`

8.1 `lattice`

8.2 `ggplot2`

Why ggplot2?

Advantages of ggplot2

consistent underlying grammar of graphics (Wilkinson, 2005)
plot specification at a high level of abstraction
very flexible
theme system for polishing plot appearance
mature and complete graphics system
many users, active mailing list

library(ggplot2)
?qplot
qplot(displ, hwy, data = mpg, colour = factor(cyl))

qplot(mpg, wt, data = mtcars)

qplot(mpg, wt, data = mtcars, colour = cyl)

qplot(mpg, wt, data = mtcars, size = cyl)

qplot(mpg, wt, data = mtcars, size = cyl, alpha = I(0.7))

qplot(mpg, wt, data = mtcars, facets = vs ~ am)

qplot(displ, hwy, data=mpg, facets = . ~ year) + geom_smooth()

p <- ggplot(mtcars)
p <- p + aes(wt, hp)
p

p + geom_point(aes(colour = factor(cyl)))

p + geom_point(aes(y = disp))

p <- ggplot(mtcars, aes(mpg, wt))
p + geom_point(colour = "darkblue")

filepath <- "http://idaejin.github.io/bcam-courses/azti-2016/introR/data/ggplot2_data.txt"

myData<-read.table(file=url(filepath),header=TRUE,sep="\t")

str(myData)

## 'data.frame':    218 obs. of  4 variables:
##  $ Tribe: Factor w/ 8 levels "Aepycerotini",..: 1 1 1 1 1 1 1 1 1 1 ...
##  $ Hab  : Factor w/ 4 levels "F","H","L","O": 3 3 3 3 3 3 3 3 3 3 ...
##  $ BM   : num  56.2 56.2 56.2 56.2 56.2 ...
##  $ var1 : num  36.5 40.9 37 36.2 36.6 37.7 37.3 39 37.7 35.3 ...

qplot(data=myData,x=BM,main="Histogram of BodyMass")

## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

qplot(data=myData,x=BM,y=var1,log="xy",color=Tribe)

8.3 Maps

Packages for Spatial Regression / Geostatistics / Spatial Point Pattern methods

sp, maptools, spatstat
maps

library(maps)

Basic syntax

map(database = "world",regions=".")

Databases are available for US, France, Italy and New Zealand. For other countries, you need to import a database with the corresponding map.

map(database = "usa")

map("state")

With the package RgoogleMaps, you can draw a background from Google Maps!

require(RgoogleMaps)

## Loading required package: RgoogleMaps

## 
## Attaching package: 'RgoogleMaps'

## The following object is masked from 'package:TeachingDemos':
## 
##     updateusr

lat <- 43.073888
lon <- -89.405236
center <- c(lat, lon)
zoom <- 18
MyMap <- GetMap(center=center, zoom=zoom)
PlotOnStaticMap(MyMap)
text(lat,lon, "X") # I missed the class!

ggmap offers plotting capabilities like ggplot2

require(ggmap)

## Loading required package: ggmap

geocode("Union South, Madison, WI")

## Information from URL : http://maps.googleapis.com/maps/api/geocode/json?address=Union%20South,%20Madison,%20WI&sensor=false

##         lon      lat
## 1 -89.29914 42.77606

Syntax	Model	Comments
`y ~ x`	\(y = \beta_0+\beta_1x\)	Straight-line with an implicit intercept
`y ~ -1 + x`	\(y = \beta_1x\)	Straight-line with no intercept; that is, a fit forced through (0,0)
`y ~ x + I(x^2)`	\(y = \beta_0+\beta_1x+\beta_2x^2\)	Polynomial model; `I()` allows for mathematical symbols
`y ~ x + z`	\(y = \beta_0+\beta_1x+\beta_2z\)	Multiple regression model
`y ~ x:z`	\(y = \beta_0+\beta_1xz\)	Model with interaction between \(x\) and \(z\)
`y ~ x*z`	\(y = \beta_0+\beta_1x+\beta_2z+\beta_3xz\)	Equivalent to `y~x+z+x:z`

Introduction to R statistical software

BCAM - Basque Center for Applied Mathematics, Applied Statistics

Dae-Jin Lee < dlee@bcamath.org >

Azti - Tecnalia

1 Introduction to the R language

1.1 Start with R

1.2 Install and load an R library

1.3 Reading data

1.4 Data Import

1.5 Exporting data

1.6 Data vectors

1.7 Basic statistics

1.8 Character vectors and factor variables

1.9 Data frames

1.10 Working with data frames

1.11 Logical vectors

1.12 Working with vectors

1.13 Matrices and arrays

1.14 Factors

1.15 Indexing vector with logicals

1.16 Missing values (NA)

1.17 Working with data frames

2 Basic data analysis in R

2.1 Basic plotting

2.2 Scatterplots

2.3 More plotting options

2.4 QQ-plot

2.5 Tables and Cross-classification

2.6 Calculation of cross-classifications

2.7 Qualitative data

2.8 Quantitative data

2.8.1 Frequency distribution of quantitative data

3 Introduction to basic programming in R

3.1 Control Structures

3.1.1 Conditional Executions

3.1.1.1 Comparison Operators

3.1.2 Logical Operators

3.2 if statements

3.3 ifelse statement

3.4 while statement

3.5 Loops

3.5.1 for

3.6 while

3.7 apply loop family

3.8 Other Loops

3.9 Improving Speed Performance of Loops

4 Probability distributions

4.1 Binomial distribution \(Bin(n,p)\)

4.2 Poisson distribution \(Pois(\lambda)\)

4.3 Aproximation of Binomial as Poisson

4.4 Exponential distribution \(Exp(\lambda)\)

4.5 The Normal distribution \(\mathcal{N}(\mu,\sigma^2)\)

4.6 Exercises

5 Statistical Inference

5.1 Interval estimation

5.1.1 Interval Estimate of Population Mean (with known variance)

5.1.2 Interval Estimate of Population Mean (with unknown variance)

5.1.2.1 Sampling Size of Population Mean

5.1.3 Interval estimates of population proportion

5.1.3.1 Sampling Size of Population proportion

5.2 Inference about two populations

5.2.1 Population mean between two matches samples

5.3 Population Mean Between Two Independent Samples

5.4 Comparison of Two Population Proportions

5.5 Goodness-of-Fit

5.6 Chi-squared Test of Independence

5.7 Mann-Whitney U test

6 Linear models in R

6.1 Simple linear regression

6.2 Defining models in R

6.3 Coefficient of Determination

6.4 Significance Test for Linear Regression

6.5 Confidence Interval for Linear Regression

6.6 Prediction Interval for Linear Regression

6.7 Residual Plot

6.8 Standardized Residual

6.9 Normal Probability Plot of Residuals

6.10 Multiple linear regression

6.10.1 Adjusted coefficient of determination

6.10.2 Significant tests and confidence/prediction intervals

Introduction to `R` statistical software

1 Introduction to the `R` language

1.1 Start with `R`

1.2 Install and load an `R` library

2 Basic data analysis in `R`

3 Introduction to basic programming in `R`

3.2 `if` statements

3.3 `ifelse` statement

3.4 `while` statement

3.5.1 `for`

3.6 `while`

3.7 `apply` loop family

6 Linear models in `R`

6.2 Defining models in `R`

8 Advanced graphics in `R`

8.1 `lattice`

8.2 `ggplot2`